LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


Class 


THE  ATMOSPHERE 

ITS  CHARACTERISTICS 
AND     DYNAMICS 


BY 

F.  J.   B.   CORDEIRO 


NEW  YORK : 
SPON   &   CHAMBERLAIN,    123   LIBERTY    STREET 

LONDON : 

E.   &  F.    N.   SPON,    LIMITED,    57   HAYMARKET,   S.  W. 

1910 


GENERAL 


COPYRIGHT,  1910 
BY   F.   J.    B.    CORDEIRO 


BURR   PRINTING   HOUSE 

18  JACOB  STREET 

NEW  YORK 


ERRATA 

On  page  46,  line  43,  should  read  ip  =  C  sec2  3,  instead  of 

-  C  sec2  3. 

On  page  47,  line  15,  should  read 


=  2  a  sin  3  1/>  32  +  R*  cos"  3  ^,.2  =  2  GO  sin  3  vr, 
in-stead  of 


-  2  K>  sn        /       32  +  R>  cos2  3  ^r2  =  2  GO  sin  3  vr. 
On  page  99,  line  28,  should  read  Fig.  24  instead  of  Fig.  25. 


BURR   PRINTING  HOUSE 

18  JACOB  STREET 

NEW  YORK 


professor  William  Xibbep 

©f  Princeton 


FOR  EARLY  ENCOURAGEMENT  IN 
THE    STUDY    OF    THE    CYCLONE 


203600 


PREFACE 


METEOROLOGY  may  be  defined  as  the  Physics  of  the  Atmosphere.  As 
such  it  is  a  purely  mathematical  science,  although  strangely  enough  its  culti- 
vators have  usually  not  been  mathematicians.  This  is  undoubtedly  the  reason 
for  its  present  backward  condition  in  comparison  with  allied  sciences. 

Among  the  exceptions  to  the  above  rule  Ferrel  stands  out  prominently. 
This  investigator,  as  far  back  as  the  fifties,  attempted  to  place  the  science 
upon  a  dynamical  basis.  His  work,  admirable  as  it  was,  seems  not  to  have 
been  generally  understood  and  often  misapprehended  by  meteorologists.  It 
was  far  from  complete,  as  must  necessarily  be  the  case  with  all  pioneer  work. 
Ferrel,  for  instance,  was  not  aware  that  a  cyclone  is  dynamically  a  gyroscope. 

It  has  for  this  reason  seemed  advisable  to  the  author  to  attempt  to  cor- 
rect and  complete  the  work  from  the  point  where  Ferrel  left  it,  and  it  is 
hoped  that  this  attempt  has  been  in  some  measure  successful.  Since  the  work 
is  necessarily  purely  mathematical,  all  methods  and  demonstrations  have 
been  presented  in  the  simplest  manner  possible,  so  that  they  may  be  under- 
stood by  the  greatest  possible  number' of  readers. 

The  general  circulation  of  the  atmosphere,  which  is  treated  at  some 
length,  has  been  found  to  be  essentially  different  from  Ferrel's  earlier  as  well 
as  his  later  conceptions  of  it.  The  mechanics  of  certain  non-meteorological 
phenomena,  such  as  sound,  and  those  coming  under  the  head  of  light,  elec- 
tricity, etc.,  have  been  explained  in  a  simple  manner. 

F.  J.  B.  C. 


BROOKLINE,  MASS.,  November  8,  1909. 


CONTENTS 


PAGE 

PREFACE v 

THE  ATMOSPHERE  : 

CONSTITUTION       .........  i 

TEMPERATURE  AND  DENSITY  .......  4 

RATE  OF  FLOW  OF  ENERGY  FROM  i  CM.2  OF  FULLY  RADIATING 

SURFACE        .........  5 

AVERAGE  TEMPERATURES         .......  6 

HEIGHT        ..........  8 

CCNVECTIVE  EQUILIBRIUM      .......  23 

VARIOUS  ATMOSPHERES  .          .          .          .          .          .          .24 

AQUEOUS  VAPOR  .........  25 

PRESSURE  IN  MMS.  OF  MERCURY  AT  SATURATION  FOR  DIFFERENT 

TEMPERATURES  OF  AQUEOUS  VAPOR                            ,         .  27 

PRESSURE     ..........  28 

PRACTICAL  BAROMETRY           .          .          .          .          .          .          .  31 

MOTION  RELATIVE  TO  THE  EARTH  : 

VERTICAL  MOTION          ........  34 

HORIZONTAL  MOTION     ........  39 

FRICTIONLESS  MOTION  OVER  A  ROTATING  SPHEROID   ...  41 

GENERAL  CIRCULATION  OF  THE  ATMOSPHERE  : 

EQUATORIAL  CIRCULATION      .......  49 

POLAR  CIRCULATION       ........  53 

MIDDLE  CIRCULATION    .          .          .         .         .         .         .          -55 

PLANETARY  CIRCULATIONS      .......  59 

ANNUAL  MEAN  PRESSURES  FOR  LATITUDE       ....  60 

CYCLONES  : 

HAMILTON'S  DESCRIPTION      .......  61 

HEARN'S  DESCRIPTION  .         .         .         .         .         .         .         .62 

FORMATION  OF  CYCLONES       .......  66 

CONVECTIONAL  HEAT  THEORY         ......  66 

DYNAMICAL  THEORY     .         .  .         .         .          .         .66 

PERPETUATION  OF  CYCLONES           ......  69 

CYCLONE  A  GYROSCOPE  ........  70 

INTRINSIC  FORCES  OF  A  CYCLONE    ......  71 

FRICTIONAL  FORCES  ACTING  ON  A  CYCLONE       ....  74 

POINSOT  CYCLONE          ........  75 

TANGENT  LAW  OF  CYCLONES          ......  76 


viii  CONTENTS 

PAGE 
DISCUSSION  OF  CYCLONES       .......         77 

REDUCTION  OF  EQUATIONS      .          .          .         .         .         .          .81 

POLAR  VELOCITY  .........         82 

FORMER  THEORIES  of  THE  MOTION  OF  A  CYCLONE    ...        84 
TORNADOES  .         .         .         .         .         .  .         .        86 

SAND  WHIRLS  AND  THUNDERSTORMS       .....         90 

OTHER  PHENOMENA  OF  THE  ATMOSPHERE: 

SOUND          ..........         91 

VELOCITY       .........         94 

POSITIVE  AND  NEGATIVE  WAVES        .....         94 

DISCONTINUITIES    .         .         .         .         .         .         .         -95 

NEWTONIAN  VALUE        .          .          .          .         .         .         .  ,      98 

WAVE  FROM  KRAKATOA  .          .          .          .         .         .         .         98 

GRAVITATIONAL  THEORY          ......        99 

OPTICAL  PHENOMENA   .          .          .          .          .          .         .         .       100 

THE  RAINBOW       .         .         .         .         .         .         .  100 

HALOES .103 

MOCK  SUNS  .          .          .         .      ^  .          ,.103 

ATMOSPHERIC  REFRACTION       ......       104 

MIRAGE          .........       106 

ELECTRICAL  PHENOMENA       .......       108 

POTENTIAL     .........       108 

LIGHTNING    .........       109 

THUNDER       .          .          .         .          .          .          .         .          .no 

AURORA          .         .         .         .         .         .         .         .         .no 

MECHANICAL  FLIGHT    .         .         .         .         .         .         .         .113 

AEROPLANES'          .         .         .         .         .         •         •         .114 

MOTORS          .         .         .         .         .         •         .        V        .       116 

HELICOPTER   .          .         .         .         .         .         .         •         .116 

APPENDIX  : 

THE  GYROSCOPE    .         .         .         .         .         .         .         .         .118 

SOME  MEMORABLE  HURRICANES      ......       126 

BAROGRAPH  OF  PORTO  RICAN  HURRICANE        .         .         .         .       128 

TABLE  OF  VELOCITIES    .         .         .         .         .         .  129 


THE 

UNIVERSITY 

OF 


THE   ATMOSPHERE 


CONSTITUTION 

THE  chief  constituents  of  the  atmosphere  are  nitrogen  and  oxygen,  and 
these  exist  in  the  approximate  proportions  by  volume  of  78  for  nitrogen  to 
21  for  oxygen.  These  proportions  are  maintained  with  great  regularity  in 
all  parts  of  the  world  and  at  all  heights  where  it  has  been  possible  to  obtain 
specimens  of  air  for  examination.  This  is  as  we  should  expect  by  reason 
of  the  diffusion  of  gases — a  rather  slow  process — as  well  as  from  the  con- 
vection due  to  the  unceasing  circulation  of  the  atmosphere.  The  proportion 
is  not  absolutely  constant,  however,  but  has  been  found  to  vary  for  oxygen 
from  20.84  to  20.97  volumes  in  a  hundred,  the  proportion  seeming  to  de- 
crease slightly  with  the  height. 

Carbonic  dioxide  is  found  everywhere  in  the  lower  regions  of  the 
atmosphere  in  the  average  proportion  of  3  parts  in  10,000  volumes.  This 
proportion  may  be  increased  enormously  in  closed  spaces  inhabited  by  ani- 
mals, as  well  as  in  volcanic  regions,  especially  where  there  are  limestone  and 
coal-bearing  strata.  This  gas  is  absorbed  by  plants  in  building  up  their  struc- 
tures, and  in  doing  so  they  fix  the  carbon  and  liberate  the  oxygen  again, 
usually  volume  for  volume  of  the  carbonic  dioxide.  Animals,  on  the  con- 
trary, inhale  oxygen  and  exhale  carbonic  dioxide  from  their  tissues,  but  there 
is  of  necessity  no  balance  between  these  two  processes  in  plants  and  animals, 
as  has  sometimes  been  supposed.  The  total  amount  of  CO2  in  the  atmos- 
phere at  any  one  period  is  too  vast  to  be  sensibly  affected  by  such  inter- 
changes. On  the  other  hand,  it  is  not  to  be  supposed  that  the  composition  of 
the  atmosphere  has  not  changed  with  the  ages,  and  there  seems  to  be  good 
reason  for  believing  that  in  times  past  the  proportion  of  CO2  was  much 
greater  than  at  present. 

Ammonia  in  minute  traces  is  found  in  the  atmosphere  and  is  chiefly  of 
organic  origin.  It  is  best  detected  in  rain  water,  and  after  a  shower  the  air  is 
practically  clear  of  it.  This  ammonia,  derived  from  organic  substances, 
partly  returns  to  them  to  supply  their  nitrogen.  The  amount  in  the  atmos- 
phere varies  from  .1  to  100  volumes  per  million. 

Traces  of  nitrous  and  nitric  acids  are  found,  which  probably  are  always 
manufactured  whenever  there  is  a  discharge  of  lightning.  Cavendish  was 
able  to  produce  these  substances  from  atmospheric  air  by  the  passage  of  an 
electric  spark.  In  the  atmosphere  these  acids  are  often  found  in  combina- 
tion with  the  ammonia.  Besides  nitrous  and  nitric  acids,  ozone  is  produced 
in  the  track  of  lightning,  but  being  very  unstable  its  existence  is  only  tempo- 


2  THE  ATMOSPHERE 

rary,  so  that,  according  to  Lord  Rayleigh,  it  is  not  to  be  considered  a  con- 
stituent of  the  atmosphere.  On  the  other  hand,  minute  quantities  of  H2O2, 
or  hydrogen  peroxide,  are  constantly  present. 

Sir  William  Ramsay  has  analyzed  atmospheric  air  in  the  following 
manner.  The  air  is  first  passed  through  a  tube  full  of  a  mixture  of  caustic 
soda  and  lime  to  remove  tfie  CO2,  and  then  through  a  U  tube  containing 
.strong  sulphuric  acid  to  deprive  it  of  its  water  and  ammonia.  Afterwards  it 
is  led  over  filings  of  red  hot  copper,  which  unite  with  the  oxygen,  and  the 
nitrogen  and  the  members  of  the  Helium  group,  being  inert,  pass  on. 

To  separate  the  nitrogen,  the  gases  are  passed  over  red  hot  magnesium, 
or  better,  a  mixture  of  magnesium  powder  and  lime,  which  gives  calcium. 
The  magnesium,  or  the  calcium,  unites  with  the  nitrogen,  and  the  inert  gases 
pass  on. 

To  separate  these  last  gases  from  each  other,  they  are  first  compressed 
in  a  bulb  and  cooled  to  — 185°  C.  by  being  immersed  in  liquid  air.  The 
Argon,  Krypton  and  Xenon  condense  to  a  liquid  with  the  Helium  and  Neon 
dissolved  in  it.  On  removing  the  bulb  from  the  liquid  air,  its  temperature 
rises,  and  the  Helium  and  Neon  escape  first,  mixed  with  a  large  amount  of 
Argon.  Argon  distils  next,  and  the  Krypton  and  Xenon  remain  to  the  last. 
By  repeating  this  process  of  fractional  distillation,  the  Argon,  Krypton  and 
Xenon  can  be  separated  from  each  other  and  from  the  Helium  and  Neon, 
the  two  latter  remaining  mixed  together  all  the  time.  To  separate  Neon 
from  Helium  recourse  must  be  had  to  liquid  hydrogen.  By  this  means  the 
mixture  of  Helium  and  Neon  can  be  cooled  to — 252°  C,  when  the  Neon 
becomes  solid  while  the  Helium  still  remains  gaseous.  Helium,  the  most 
refractory  of  all  gases,  has  been  liquefied.  Its  boiling  point  is  about  — 267° 
C.  or  6°  absolute. 

One  hundred  volumes  of  air  contain  only  .937  volume  of  these  inert 
gases,  and  by  far  the  larger  portion  of  this  is  Argon,  which  exists  in  the 
proportion  of  not  quite  i%  by  volume  in  the  atmosphere.  The  amount  of 
the  other  four  gases  taken  together  is  about  j-^  of  that  of  Argon. 

The  spectrum  of  the  Aurora  contains  lines  of  Argon,  Krypton  and 
Xenon,  the  Krypton  lines  being  most  clearly  and  uniformly  seen.  The  yel- 
low-green line  of  the  aurora,  which  has  long  been  known  as  characteristic  of 
it  and  only  recently  identified,  is  with  little  doubt  identical  with  the  yellow- 
green  line  of  Krypton,  their  wave  lengths  being  respectively  .5571  and. 5570^. 

Besides  these  regular  constituents  of  the  atmosphere,  certain  extraneous 
solid  substances  are  so  constantly  present  as  to  necessitate  their  being  consid- 
ered a  part  of  the  atmosphere.  Although  any  solid  substance  in  the  air  must, 
on  the  whole,  be  constantly  falling,  still  they  are  maintained  aloft  by  currents 
at  times  for  indefinite  periods.  The  dust  which  is  swept  from  the  surface 
of  the  earth  by  winds  probably  never  stays  suspended  for  any  length  of  time. 
That  which  is  shot  up  by  volcanoes  to  very  great  heights,  especially  the  im- 


CONSTITUTION  3 

palpable  dust  resulting  from  the  condensation  of  volcanic  gases  at  great 
heights  and  low  temperatures  (SO2),  may  remain  suspended  for  much  longer 
periods.  But  the  constant  rain  of  meteoric  bodies,  which  though  mostly  of 
microscopic  dimensions,  are  moving  at  the  rate  of  about  twenty  miles  a  second 
when  they  meet  the  atmosphere  and  are  consequently  sublimated  at  the 
highest  levels,  keeps  the  atmosphere  as  a  whole  supplied  with  a  constant  pro- 
portion of  falling  solid  constituents.  Although  this  cosmic  dust,  the  particles 
of  which  may  vary  from  I  mm.  to  .001  mm.  in  diameter,  is  relatively  small, 
still  in  the  aggregate  it  is  considerable.  It  is  sufficient  to  color  freshly  fallen 
snow  on  mountain  tops  and  in  the  polar  regions.  It  is  composed  principally 
of  iron  with  traces  of  nickel  and  cobalt — all  paramagnetic  substances.  Such 
dust  is  easily  swept  up  with  a  magnet,  and  in  fact  the  greater  amount  of  it  in 
the  polar  regions  than  elsewhere  is  due  to  the  magnetic  attraction  of  the 
earth's  poles.  It  is  probably  effective  in  intensifying  auroral  and  other  mag- 
netic phenomena  and  also,  in  conjunction  with  oxygen,  which  is  also  a 
paramagnetic  substance,  in  increasing  the  permeability  of  the  atmosphere. 

Lastly  we  have  to  consider  the  presence  of  aqueous  vapor.  This  gas  is 
always  present  near  the  surface  of  the  earth,  though  in  exceedingly  variable 
proportions,  while  at  very  moderate  levels  it  is  almost  entirely  lacking.  We 
shall  have  much  to  say  of  this  very  important  constituent.  In  fact,  from  all 
points  of  view,  the  three  gases  of  practical  importance  in  the  atmosphere  are 
oxygen,  nitrogen  and  aqueous  vapor. 

The  constituents  of  the  atmosphere  are,  therefore, 


1 

7ols.  in     Atomic 
ico          Weight 

Nitrogen  .  . 

78            14    Liquefies  at  —  194°  C. 

Solidifies  —  214°  C. 

Oxygen  .  .  . 

21     15.882     Liquefies  at  —  180.5°  C, 

Argon  .... 

i         39.9     Liquefies  at  —  186.9°  C. 

Solidifies  —  189.6' 

CO2  

.0^                     Liquefies  at  —  Q1;.^0  C. 

Solidifies  —  107.5° 

vij                                  -"T,  *-*  ^  *  *  v*  tj   «•)«)          yj    *j       v-/» 

Krypton  .  . 

Trace     81.5 

Xenon  .... 

128 

Neon  

20 

Helium  .  .  . 

4    Liquefies  at  —  267°  C. 

Hydrogen  . 

"       1.007     Liquefies  at  —  252°  C. 

Solidifies  —258°  C. 

SO2  

Near  volcanoes. 

H202  

H2NO2... 

" 

H2NO3  .  .  . 

if 

NH3  

" 

HnO 

Variable.     From  a  trace  to  7  .  t:  volumes 

C. 


Besides  which  the  following  solid  substances  are  usually  found  floating 
in  the  atmosphere,  all  in  minute  traces. 


4  THE  ATMOSPHERE 

Iron       1  Calcium  Carbonate  "| 

Nickel    V  Cosmics.  Magnesium  Carbonate  ^,         ., 

r  u  u    f  c-r  From  the 

Cobalt    1  Silica  c         ( 

t  .  y  surface  of 

Aluminum  _,  ,, 

XT  n      T7         .u  AT    c^  the  earth. 

NaCl.     From  the  sea.  Na2SO4 

Organic  matter  and  living  germs  J 

TEMPERA  TURE  AND  DENSITY 

IT  is  easy  to  calculate  the  total  mass  of  the  atmosphere,  as  was  done  long 
ago  by  Pascal,  but  such  huge  figures  have  little  practical  significance.  If  we 
say  that  in  round  numbers  the  total  amount  of  air  attached  to  the  earth  is 
5,500  million  million  tons,  we  derive  little  benefit  from  this  information.  Of 
more  importance  is  the  determination  of  its  height,  its  shape  and  its  char- 
acteristics at  different  levels.  To  do  this  we  must  be  guided  by  the  follow- 
ing laws,  which  have  been  derived  from  experiment.  The  principal  law 
which  applies  to  all  perfect  gases  is  expressed  by  the  formula  p  v  =  R  # 
where  p  is  the  pressure,  v  the  volume,  #  the  temperature,  and  R  a  constant 
depending  upon  the  units  selected.  This  is  called  the  characteristic  equation 
of  a  perfect  gas  and,  although  when  far  removed  from  their  condensation 
points,  gases  follow  this  formula  rather  closely,  still  no  gas  does  so  exactly. 
In  other  words,  no  gas  is  a  perfect  gas.  By  a  convenient,  though  strictly 
improper  use  of  the  term,  a  gas  is  spoken  of  as  being  more  or  less  perfect 
according  as  it  follows  this  formula  more  or  less  closely. 

The  air,  like  all  material  substances,  is  subject  to  the  attraction  of  the 
earth;  consequently  the  particles  at  any  level  are  not  only  subjected  to  this 
force,  but  are  also  pressed  by  the  superincumbent  weight  of  all  the  higher 
layers.  The  pressure  is,  therefore,  a  maximum  at  the  surface  of  the  earth 
and  zero  at  the  superior  limit.  To  determine  the  density  of  the  air  at  any 
particular  point,  it  is  not  only  necessary  to  know  the  pressure,  but  also  the 
temperature  at  that  point,  and  since  the  temperature  of  the  atmosphere  is  due 
practically  entirely  to  the  sun,  the  amount  of  heat  received  from  the  earth 
being  infinitesimal  by  comparison,  we  shall  begin  our  study  of  temperatures 
with  the  sun. 

With  regard  to  radiant  heat,  it  has  been  found  that  the  amount  of  such 
heat  absorbed  or  emitted  by  a  body  depends  greatly  .upon  the  nature  of  the 
surface  bounding  it.  If  such  a  surface  reflects  a  large  part  of  the  heat  falling 
upon  it,  whether  from  the  outside  or  inside,  it  will  absorb  or  emit  a  corre- 
spondingly lesser  proportion  of  this  heat.  A  surface  thickly  coated  with  lamp 
black  reflects,  regularly  and  irregularly,  an  insignificant  amount  of  the  radi- 
ant energy  falling  upon  it  and  is  taken  as  a  practical  realization  of  a  fully 
absorbing  or  fully  radiating  substance.  Such  a  substance  has  been  called  a 
"black  body,"  but  since  under  the  influence  of  intense  heat  it  may  become 
white  hot,  it  seems  better  to  refer  to  it  as  a  "fully  radiating  substance." 


TEMPERATURE  AND  DENSITY  5 

For  a  long  time  it  was  sought  to  find  a  law  connecting  the  outflow  of 
energy  from  a  fully  radiating  body  with  its  temperature.  Stefan  was  the 
first  to  suggest  a  formula  which  agreed  at  all  satisfactorily  with  experimental 
results,  and  we  shall  call  this  formula  Stefan's  Law.  It  is  this — that  the 
stream  of  energy,  or  amount  of  energy  radiated  in  a  second  from  a  fully 
radiating  body,  is  proportional  to  the  fourth  power  of  its  temperature  reck- 
oned from  the  absolute  zero.  Experimental  determinations  of  the  stream 
of  energy  issuing  from  a  fully  radiating  body  were  first  carried  out  by 
Pouillet  and  Professor  Herschel. 

In  determining  the  amount  of  energy  received  from  the  sun  at  the  sur- 
face of  the  earth,  there  are  great  difficulties,  chief  of  which  is  the  calculation 
of  the  amount  which  is  absorbed  and  reflected  by  the  layers  which  the  sun's 
rays  have  traversed.  We  may,  however,  assume  that  we  are  not  far  from 
the  truth  when  we  take  the  amount  of  energy  from  the  sun  falling  vertically 
upon  one  square  centimeter  of  a  fully  absorbing  surface  just  outside  the 
earth's  atmosphere,  as  £%  of  a  calorie  per  second.  That  is  to  say,  if  we 
had  a  cube  each  face  of  which  was  one  square  centimeter  and  one  of  these 
faces  was  blackened  and  exposed  perpendicularly  to  the  sun's  rays,  and  if 
the  cube  were  filled  with  water,  its  temperature  would  rise  at  first  •£%  of  a 
degree  C.  every  second.  Now  the  area  of  a  sphere  around  the  sun  at  the 
distance  of  the  earth  is  46,000  times  the  area  of  the  sun's  surface.  The 
stream  of  energy  from  one  square  centimeter  of  the  sun's  surface  is,  there- 
fore, 46,000  X  ?-%  calories,  or  1920  calories  per  second.  . 

Kurlbaum,  who  has  made  determinations  of  the  amount  of  energy 
issuing  from  a  fully  radiating  surface,  has  prepared  the  following  table : 

RATE  OF  FLOW  OF  ENERGY  FROM  i  CM.2  OF  FULLY  RADIATING 

SURFACE. 

Grammes  of  Water  Heated  i°  per 


Absolute  Temperature 

0° 

100° 

300° 

IOOO° 

3OOO0 

6000° 

6250° 


second,  or  Calories 
O.OOOOOO 
0.000127 
O.OIO3OO 
I . 27OOOO 
IO3.OOOOOO 
I65O.OOOOOO 
I930.OOOOOO 


We  see  from  this  table  that  the  amount  of  energy  streaming  from  the 
sun's  surface  corresponds  approximately  to  a  temperature  of  6250°  absolute 
or  about  6000°  C.  Now,  if  we  suppose  a  sphere  of  I  cm.2  cross-section  fur- 
nished with  a  fully  absorbing  surface  at  the  distance  of  the  earth,  such  a 
sphere  will  have  4  cm.2  of  surface.  It  is  receiving  from  the  sun  ^  calorie 
every  second,  and  when  it  has  arrived  at  a  state  of  equilibrium  it  will  radi- 
ate every  second  exactly  this  amount.  Consequently  the  radiation  per  cm.2 


THE  ATMOSPHERE 


will  be  V^  °f  a  calorie,  or  .0104  calorie  per  second.  From  Kurlbaum's  table 
we  see  that  this  corresponds  to  a  temperature  of  300°  absolute,  or  27°  C. 
The  average  temperature  of  the  surface  of  the  earth,  as  usually  estimated, 
is  16°  C.,  so  that  if  we  suppose  that  about  11°  has  been  lost  through  absorp- 
tion and  reflection  from  the  upper  layers  of  the  atmosphere,  we  find  here  a 
tolerable  confirmation  of  our  calculation.  Langley,  from  his  bolometric 
measurements,  has  estimated  that  41%  of  the  sun's  energy  fails  to  reach  the 
surface  of  the  earth  directly.  It  has  also  been  found  that  the  radiation  from 
the  sun  is  not  constant  (Langley  and  Abbott),  but  is  subject  to  fluctuations — 
amounting  at  times  to  possibly  20%  of  the  total. 

The  temperature  at  the  earth's  surface  has  been  measured  at  so  many 
times  and  places  that  we  can  give  very  accurate  averages  for  many  places, 
and  in  fact  have  a  very  good  idea  of  the  distribution  of  temperatures  over 
the  whole  globe.  These  temperatures  vary  greatly,  of  course,  with  place 
and  season,  but  at  any  one  place  the  average  for  a  year  remains  remarkably 
constant. 

The  highest  known  temperature  of  the  air  (in  shade)  was  taken  at 
Murzouk,  in  Africa,  and  was  56.4°  C.  But  a  thermometer  buried  in  the  soil 
in  Africa  has  given  74°  C.  Among  the  greatest  degrees  of  cold  ever  regis- 
tered is  — 60°  C.  at  Semipalatinsk  in  Siberia,  and  — 56.5°  C.  at  Fort  Re- 
liance, in  North  America.  Captain  Nares  recorded  a  temperature  of  — 65°  C. 
The  following  table,  compiled  from  Buchan's  isothermal  charts,  will  give  a 
good  idea  of  the  average  temperature  at  various  latitudes. 

TEMPERATURE  CENTIGRADE, 


Latitude 

January 

July 

Mean  of  Year 

+80 

-3T-9 

+  1.0 

—15-5 

70 

—26.5 

6.9 

-9.8 

60 

—  16.9 

13-8 

—  1.6 

50 

—  6.0 

18.6 

+  6.3 

40 

+  4-5 

22.8 

13.6 

30 

12.9 

26.6 

19.8 

20 

21.7 

29.0 

25-3 

10 

25-9 

28.4 

27.2 

0 

27-3 

26.  i 

26.7 

—  10 

29.9 

24.0 

25-9 

20 

26.6 

20.8 

23-7 

30 

23.0 

15.6 

19-3 

40 

17.6 

II.  I 

14.4 

50 

ii  .  i 

6.4 

8.8 

60 

3-6 

0.0 

1.8 

TEMPERATURE  AND  DENSITY  7 

According  to  more  recent  determinations  by  Hann,  the  average  tem- 
peratures in  high  southern  latitudes  are  somewhat  less  than  those  given 
above.  As  we  have  said  before,  the  average  temperature  of  the  whole  sur- 
face of  the  earth  is  about  16°  C. 

While  our  knowledge  of  temperatures  at  the  surface  of  the  earth  is  so 
definite,  we  have  as  yet  only  general  ideas  as  to  what  it  is  at  higher  levels. 
It  is  easily  noted  that  the  temperature  on  the  whole  falls  as  we  ascend — 
perhaps  something  like  .4°  C.  for  each  hundred  meters  in  the  lower  layers. 
But  there  is  no  regular  law,  and  the  rate  of  fall  becomes  less  as  we  ascend. 
There  is  at  times  an  "inversion"  of  temperature,  noticeable  especially  at 
night,  when  owing  to  the  rapid  radiation  from  the  earth's  surface,  the  layers 
of  air  in  close  proximity  to  the  earth  are  much  colder  than  those  above.  It 
may  happen  in  balloon  ascensions  that  the  passage  is  made  through  alternate 
layers  of  hotter  and  colder  strata. 

The  thermometry  of  the  upper  atmosphere  is  attended  with  considerable 
difficulties  owing  to  the  disturbance  produced  by  radiation,  either  directly 
from  the  sun  or  from  surrounding  objects.  Thus,  although  the  air  itself  has 
a  very  definite  temperature,  which  falls  considerably  as  we  attain  to  great 
heights,  yet  a  fully  absorbing  body  exposed  to  the  rays  of  the  sun  would 
increase  its  temperature  until  at  the  upper  limit  it  would  acquire  a  tempera- 
ture of  27°  C.,  while  the  surrounding  air,  what  was  left  of  it,  would  be 
somewhere  near  the  absolute  zero.  A  climber  on  a  lofty  mountain  may  suffer 
from  the  heat  when  exposed  to  the  rays  of  the  sun,  although  he  is  walking 
on  ice  and  the  actual  temperature  of  the  air  is  below  freezing.  For  this 
reason  the  earlier  data  derived  from  balloon  ascensions  were  valueless  be- 
cause, even  though  screened  from  the  direct  rays  of  the  sun,  they  were  not 
protected  from  the  radiation  of  the  balloon  itself,  which  sometimes  became 
much  hotter  than  the  surrounding  air.  Probably  the  best  way  of  taking 
such  readings  would  be  to  enclose  the  thermometer  in  a  vessel  with  double 
sides,  having  a  vacuum  between  them,  and  the  sides  silvered,  like  the  recep- 
tacles in  which  liquid  air  is  carried. 

Great  heights  have  been  reached  by  manned  balloons.  Dr.  Berson,  of 
Berlin,  ascended  to  30,000  ft.  in  1894.  The  barometer  was  nine  inches  and 
the  temperature  — 48°  C. 

Dr.  Berson  and  Professor  Suring,  in  1901,  attained  to  a  height  of  10,300 
meters  (nearly  6^  miles).  The  recorded  temperature  was  — 40°  C.  This 
is  the  greatest  authentic  height  ever  reached  by  man.  Glaisher  and  Coxwell, 
in  1862,  became  unconscious  at  29,000  ft.,  after  which  the  balloon  was  sup- 
posed to  have  risen  to  36,000  ft.,  but  this  is  uncertain.  Berson  and  Suring 
remained  conscious  by  inhaling  oxygen. 

Within  recent  years  considerable  progress  has  been  made  in  obtaining 
temperatures  at  great  heights  by  means  of  kites  (Rotch)  and  sounding  bal- 
loons (Ballon-Sondes) — small  balloons  to  which  are  attached  automatically 


8  THE  ATMOSPHERE 

registering  instruments.  A  German  balloon  of  this  type,  called  the  Cirrus, 
on  one  occasion  rose  to  a  height  of  54,000  ft.,  and  registered  a  minimum 
temperature  of  — 52.5°  C.  On  another  occasion  it  rose  to  a  height  of 
61,000  ft.  with  a  minimum  temperature  of  — 61°  C.,  and  on  still  another 
occasion  it  rose  to  72,000  ft.  (134  miles)  and  brought  down  a  temperature 
of-35°C. 

A  balloon  released  at  Tromso,  Norway,  reached  a  height  of  ten  miles, 
registering  a  temperature  of  — 60°  C.  at  a  little  over  six  miles,  and  a  tem- 
perature of  — 47°  C.  at  ten  miles. 

Finally,  a  balloon  released  at  Uccle  in  Belgium,  in  the  winter  of  1908, 
rose  to  a  height  of  95,250  ft.,  or  eighteen  miles,  recording  a  minimum  pres- 
sure of  two-fifths  inches  of  mercury  and  a  temperature  of  — 67°  C.  at  eight 
miles,  and  a  temperature  of  — 64°  C.  at  eighteen  miles. 

From  the  fact  that  balloons  have  frequently  registered  a  greater  tem- 
perature at  very  great  heights  than  at  intermediate  points,  it  has  been  sup- 
posed that  there  is  a  permanent  inversive  layer  at  those  heights  where  the 
temperature  after  steadily  falling  begins  to  rise  again  (Hergesell,  Arrhenius). 
But  the  fact  that  no  a  priori  reason  can  be  adduced  for  such  a  phenomenon 
renders  it  extremely  doubtful,  if  not  improbable.  The  effect  of  radiation 
vitiates  somewhat  all  these  measurements.  A  balloon  exposed  to  the  rays 
of  the  sun  after  it  has  risen  through  the  greater  part  of  the  atmosphere  and 
completely  through  the  blanket  of  aqueous  vapor,  which  possesses  the 
strongest  absorptive  power,  is  practically  unscreened.  The  balloon  itself,  no 
matter  what  its  height,  becomes  quite  hot  under  these  conditions,  and  will 
exert  an  influence  on  its  thermometers  even  though  they  are  screened.  While 
it  is  possible  that  such  a  permanent  inversive  layer  exists  at  great  heights, 
considerable  more  proof  will  have  to  be  adduced  than  is  at  present  at  hand 
for  its  acceptance. 

From  the  soundings  which  have  already  been  taken  we  can  get  some- 
thing of  an  idea  of  the  distribution  of  the  average  isothermal  surfaces  in  the 
upper  air,  but  much  remains  to  be  done  before  our  knowledge  can  in  any 
sense  be  considered  accurate. 

Fig.  i  gives  a  rough  indication  of  this  distribution.  The  temperatures 
are  Centigrade  and  the  heights  above  the  earth's  surface  are  given  in  miles. 

HEIGHT 

'  The  height  to  which  the  atmosphere  extends  has  been  the  subject 
of  much  speculation  and  various  estimates.  That  there  is  an  upper 
limit  is  evident  from  various  considerations.  It  is  not  necessary  to  suppose, 
as  has  been  done,  that  after  reaching  a  certain  degree  of  rarefaction  a  gas 
loses  all  power  of  expansion  and  becomes  as  it  were  an  extremely  tenuous 
liquid.  With  our  best  air  pumps,  which  reduce  the  pressure  to  a  small 


HEIGHT  9 

fraction  of  a  millimeter,  it  is  always  possible  to  exhaust  a  little  more,  and 
there  is  no  evidence  of  a  limit  being  approached.  There  is,  however,  a  uni- 
versal ether  which  pervades  all  space  and  which,  being  a  material  substance, 
is  supposed  to  have  a  definite  density.  We  know  very  little  about  this  ether, 
and  some  very  vague  and  rather  contradictory  views  have  been  entertained 
regarding  it.  As  it  carries  all  radiations,  serving  as  a  medium  of  communica- 
tion between  distant  bodies,  it  seems  certain  that  it  is  a  material  substance 


FIG.  i 

endowed  with  the  properties  of  elasticity  and  inertia  of  ordinary  matter.  It 
must  also  be  the  medium  through  which  attraction  (or  repulsion)  is  exerted 
between  two  bodies,  unless  we  suppose  an  independent  medium  for  this  pur- 
pose, which  is  not  at  all  logical.  Graetz  has  estimated  the  density  of  the 
ether  as  9  X  io~16  that  of  water.  Lord  Kelvin,  while  giving  no  definite 
value,  has  calculated  that  it  must  be  greater  than  io~18. 

Another  possible  estimate  may  be  arrived  at  as  follows.     The  velocity 
of  a  disturbance  in  this  medium,  which  is  the  velocity  of  all  radiations,  viz., 

300,000  kilometers  per  second,  should  be  \f  -p,  where  E  is  its  elasticity  and 

D  its  density.  Now  the  elasticity  or  pressure  of  this  medium  may  possibly 
be  inferred  from  magnetic  attraction.  If,  as  seems  probable,  magnetic  lines 
of  force  are  whirls  in  the  ether  which  by  centrifugal  force  produce  a  partial 
ether  vacuum  along  their  axes,  just  as  a  vortex  in  a  fluid  produces  a  partial 
vacuum,  then  the  attraction  between  two  magnets  is  to  be  explained  as  the 


27,577,6°7 


10  THE  ATMOSPHERE 

general  pressure  of  the  ether  tending  to  destroy  this  vacuum.  The  greatest 
magnetic  force  which  it  has  been  possible  to  produce  is  about  400  Ibs.  to  the 
square  inch,  which  we  may  suppose  to  be  an  approach  to  a  perfect  ether 
vacuum,  and  consequently  to  indicate  the  general  pressure  of  the  ether. 

Now  400  Ibs  to  the  square  inch  is  equivalent  to  27,577,607  dynes  per 

/27  <577 
square  centimeter.    Consequently  we  may  write  *3  X  io10  =   \  / -FT 

where  our  values,  being  expressed  in  centimeters  and  grammes,  will  give  the 
density  in  grammes  per  cubic  centimeter. 

This  gives  us  somewhat  less  than  J  X  io~14  grammes  per  c.c.,  or  in 
round  numbers  the  density  of  the  ether  is  \  X  io~~14  that  of  water. 

It  must  be  remembered  that  all  this  is  highly  hypothetical  and  scarcely 
susceptible  of  proof.  But  whether  we  take  Graetz's  estimate  of  roughly 
IO~17  times  that  of  water,  or  our  present  estimate  of  ^  X  io~14,  it  is 
certain  that  the  air  must  attain  this  extreme  rarity  at  a  comparatively  short 
distance  from  the  earth.  It  is  difficult  to  conceive  any  atmosphere  less  dense 
than  the  ether  as  belonging  to  the  earth,  and  apart  from  this  there  is  some 
reason  for  thinking  that  if  gravitation  is  effected  through  the  ether,  then  any 
matter  less  dense  than  this  medium  would  be  repelled  instead  of  attracted 
by  a  heavy  body.f 

Gravitation  must  be  effected  through  some  medium  filling  the  space 
between  two  distant  bodies,  for  if  this  space  were  an  absolute  vacuum  no 
interaction  of  any  kind  could  take  place  between  them.  As  we  have  said 
before,  having  already  one  medium,  viz.,  the  ether,  it  would  be  illogical  to 
postulate  still  another  special  medium  by  which  gravitation  is  effected.  We 
have  been  forced  to  consider  the  ether  a  material  substance  having  at  least 
two  of  the  properties  of  ordinary  matter.  It  must  possess  elasticity  and 
inertia  in  order  to  transmit  waves,  though  we  cannot  conceive  it  to  have 
weight  or  temperature.  We  can  conceive  it  to  be  full  of  waves  of  radiational 
(gravitational)  energy,  but  it  is  the  bearer  of  this  energy  merely.  We  can- 
not well  suppose  that  it  has  any  temperature  itself  or  that  it  exerts  of  itself 
any  attractional  or  repulsional  force  on  other  bodies  or  on  itself  or  is  at- 
tracted or  repelled  by  them.  Now  it  can  be  shown,  both  theoretically  and 
experimentally,  that  when  a  heavy  body  sends  waves  through  an  elastic 
medium,  all  bodies  denser  than  the  medium  which  are  swept  by  these  waves 
will  be  attracted,  while  less  dense  bodies  will  be  carried  along  with  the  waves,, 
the  limiting  velocity  in  this  case  being  the  velocity  of  wave  propagation, 
(v.  Article  Gravitation.  1.  c.) 

According  to  this  view,  when  the  air  has  reached  a  limiting  density  of 
^  X  io~14,  or  the  density  of  the  ether,  it  would  no  longer  be  attracted  by 


*  3  X  io10  centimeters  per  second  is  the  velocity  of  light. 

t  See  Article  Gravitation,  Popular  Astronomy,  January,  1905. 


HEIGHT  n 

the  earth  but  would  be  driven  off  with  a  high  velocity.  Owing  to  the  extreme 
tenuity  of  the  air  at  this  limit,  the  rate  of  loss  is  very  small  and  it  can  be 
easily  calculated  that  it  would  take  many  millions  of  years  for  a  complete 
dissipation  of  the  atmosphere.  That  the  earth  is  continually  losing  minute 
quantities  of  its  atmosphere  seems  quite  probable  judging  from  the  analogy 
of  other  planets.  Thus  the  amount  of  the  atmosphere  surrounding  the  earth 
is  considerably  greater  than  that  of  Mars  or  the  Moon,  the  latter  seeming  to 
have  lost  nearly  all,  while  on  the  other  hand  it  is  less  than  that  of  Venus  or 
the  outermost  planets,  (v.  Article  The  Atmosphere,  Popular  Astronomy, 
August,  1905.)  When  a  wave  sweeps  over  an  aggregation  of  molecules,  por- 
tions of  the  wave  will  be  mutually  reflected  among  them,  some  of  the  mole- 
cules being  shielded  by  others  from  the  wave.  For  a  sufficient  depth  of  the 
wave,  the  projections  of  the  molecules  upon  a  section  perpendicular  to  the 
wave  would  form  a  practically  continuous  surface.  To  be  added  to  this  are 
the  mutual  interactions  between  the  molecules  or  their  resistance  to  being 
crowded  together.  For  all  these  reasons,  therefore,  given  a  certain  distribu- 
tion of  molecules  to  volume,  the  action  of  the  wave  on  them  will  be  the  same 
as  on  a  simple  body  distributed  continuously  over  the  same  space  with  a 
uniform  density.  This  uniform  density  will  be  the  quotient  of  the  sum  of  the 
masses  of  all  the  molecules  divided  by  the  space. 

There  is  considerable  difficulty  in  accepting  the  modern  theory  of  radia- 
tion pressure.  That  light  exerts  pressure  is  experimentally  demonstrated  by 
causing  it  to  strike  upon  a  light  vane  in  a  vacuum,  in  which  case  the  vane 
is  repelled  (Lebedew,  Nichols).  The  experiment  proves  beyond  a  doubt 
that  the  ether  has  both  mass  (inertia)  and  elasticity  and  hence  is  a  material 
substance,  but  beyond  this  proves  very  little.  That  the  ethereal  waves  falling 
upon  a  surface  much  larger  than  the  waves  and  being  reflected  from  it  should 
exert  pressure,  was  to  be  expected  if  the  ether  is  a  material  substance. 

The  theory  supposes  that  the  pressure  is  simply  proportional  to  the 
surface  exposed,  while  the  gravitational  force  which  is  taken  as  always 
attractive  is  proportional  to  the  mass.  As  for  the  same  substance  the  mass 
is  as  the  cube  of  its  linear  dimensions,  while  the  surface  is  as  the  square,  it 
follows  that  the  repellent  pressure  is  to  the  gravitational  attraction  inversely 
as  the  linear  dimensions.  Hence,  by  reducing  the  size  of  the  body  a  limit 
will  be  reached  where  the  repulsion  is  counterbalanced  exactly  by  the  attrac- 
tion. If  the  body  becomes  still  smaller  the  light  pressure  will  overcome  the 
force  of  attraction  and  the  body  will  be  driven  off.  It  is  calculated  that  a 
sphere  having  a  diameter  of  nnnnnr  °f  a"  mch  would  be  driven  away  from 
the  sun  by  its  light,  which  in  this  case  is  more  powerful  than  the  gravitational 
attraction.  Whether  in  case  the  sphere  were  a  "black  body,"  and,  therefore, 
did  not  reflect  any  light,  the  results  would  be  changed,  has  not  been  con- 
sidered. The  theory  is  objectionable  in  that  it  postulates  a  double  set  of 
forces,  one  a  repulsional  force  based  upon  an  experiment  in  reflection,  while 


12  THE  ATMOSPHERE 

the  force  of  gravitation  remains  as  mysterious  and  unexplained  as  before 
and  necessarily  having  no  connection  with  ethereal  vibrations. 

It  is  further  untenable  from  the  fact  that  the  molecules  in  the  atmos- 
pheres of  both  the  sun  and  the  earth  are  far  below  the  limit  where  the 
radiation  pressure  exceeds  the  gravitational  attraction.  Hence,  were  the 
theory  true,  the  atmospheres  of  all  bodies  would  immediately  fly  off,  leaving 
them  bare. 

The  attempt,  therefore,  to  explain  repulsional  phenomena  (comets'  tails) 
as  a  differential  effect  between  radiation  pressure  and  gravitational  attraction 
fails.  Rather  does  it  seem  more  consistent  with  facts  and  more  supportable 
by  theory,  instead  of  multiplying  actions,  to  find  in  the  single  action  of 
gravitation  alone  the  force  which  either  attracts  or  repels  according  as  the 
body  acted  upon  is  denser  or  rarer  than  the  medium  which  carries  the  action. 

In  this  connection  we  may  briefly  mention  the  theory  of  Dr.  G.  Johnstone 
Stoney,  by  which  he  attempts  to  account  for  the  absence  of  hydrogen  in  the 
atmosphere.  It  is  supposed,  though  there  is  no  direct  reason  for  thinking  so, 
that  the  atmosphere  formerly  must  have  contained  much  more  hydrogen 
than  at  present.  It  is  now  a  mere  trace.  According  to  the  kinetic  theory,  a 
hydrogen  molecule  is  supposed  to  have  an  average  velocity  of  i|  miles  per 
second  at  o°  C.  A  molecule  of  oxygen  at  the  same  temperature  is  supposed 
to  have  an  average  velocity  of  rather  less  than  one-third  mile  per  second,  and 
so  on  for  the  other  gases,  the  velocity  decreasing  as  the  density  increases. 
But  these  are  merely  the  average  velocities :  some  molecules  are  supposed  to 
move  faster,  others  slower,  and  a  considerable  proportion  of  hydrogen 
molecules  may  be  conceived  as  having  a  higher  velocity  than  seven  miles  per 
second.  Now,  this  velocity,  directed  away  from  the  earth,  is  sufficient  to 
carry  a  body  beyond  the  attraction  of  the  earth  into  space,  provided  it  is  at 
the  upper  limit  of  the  atmosphere.  During  the  lapse  of  ages,  Dr.  Stoney 
supposes  that  in  this  way  a  considerable  mass  of  hydrogen,  practically  all 
that  the  earth  once  possessed,  has  been  lost  from  our  atmosphere,  as  well  as 
a  lesser  proportion  of  oxygen  and  nitrogen,  etc.  But  it  seems  to  have  been 
forgotten  that  at  the  upper  limit  of  the  atmosphere  the  temperature  falls 
nearly  to  that  of  space,  which  is  probably  not  far  from  the  absolute  zero. 
Thus  at  a  very  short  distance  from  the  earth's  surface  the  average  velocity 
of  a  hydrogen  molecule  would  be,  not  i^  miles  a  second,  but  a  minute  frac- 
tion of  this  amount,  and  the  proportion  of  molecules  having  a  velocity  of 
S£ven  miles  would  be  infinitesimally  small.  The  most  probable  supposition  is 
that  at  the  upper  surface  of  the  atmosphere  the  temperature  is  actually  the 
absolute  zero,  so  that  it  would  be  impossible  for  any  gas  to  escape  actively 
from  the  earth.  As  we  have  already  pointed  out,  it  is  very  probable  that 
there  is  a  dissipation  of  the  atmosphere,  but  the  process  is  a  passive  one,  the 
gases  being  driven  off  by  the  action  of  the  earth. 

We  have  assumed  a  natural  limit  for  the  atmosphere,  beyond  which  it 


HEIGHT  13 

cannot  extend,  viz.,  the  point  where  its  density  becomes  equal  to  that  of  the 
ether.  Another  limit  might  possibly  suggest  itself,  viz.,  the  point  where 
nitrogen  freezes,  — 214°  C.,  and  any  air  existing  at  such  a  level  might  be 
supposed  to  be  solid.  Although,  as  we  shall  see  later  on,  there  is  practically 
such  a  limit  for  the  aqueous  vapor  of  the  atmosphere,  none  exists  for  the 
other  gases.  According  to  Dewar,  the  maximum  tension  of  gaseous  air  at 
35°  Absolute,  which  is  the  boiling  point  of  hydrogen,  is  somewhat  less  than 
.002  mm.  of  mercury.  Of  course,  at  this  temperature  and  at  ordinary  pres- 
sures air  is  frozen  solid,  but  a  portion  can  always  exist  in  the  gaseous  state 
having  a  certain  maximum  pressure  for  each  temperature.  A  slight  increase 
in  pressure  will  condense  some  of  the  gas,  and  this  saturation  temperature 
of  a  gas  is  called  the  Dew  Point.  Thus  at  — 32°  C.  aqueous  vapor  has  a 
maximum  pressure  of  .32  mm.  of  mercury.  The  pressure  at  all  points  in 
the  atmosphere,  however,  is  very  much  less  than  the  maximum  pressure  cor- 
responding to  the  temperature;  consequently,  the  ordinary  gases  can  never 
become  solid  or  liquid  in  the  atmosphere. 

Owing  to  centrifugal  force,  due  to  the  earth's  rotation,  a  fluid  envelope 
must  assume  the  form  of  a  spheroid.  A  surface  of  water  covering  the  earth 
would  have  an  equatorial  radius  greater  than  the  polar  radius  by  something 
over  thirteen  miles,  or  about  the  ellipticity  of  the  earth.  The  surface  of  the 
atmosphere  where  it  touches  the  earth  has,  of  course,  a  like  ellipticity,  but 
that  of  the  upper  surface  is  greater.  It  would  be  easy  to  calculate  it  pro- 
vided the  temperature  were  constant  throughout.  But  the  temperature  being 
less  at  the  poles  than  at  the  equator,  the  ellipticity  of  the  upper  surface  is 
somewhat  greater  than  with  a  uniform  temperature.  Actually  the  upper 
surface  of  the  atmosphere,  although  approximating  to  a  spheroid,  is  far 
from  being  a  mathematically  smooth  surface,  being  in  fact  rough  and 
irregular.  We  have  probably  in  the  sun  an  analogue  which  the  earth 
repeats  on  a  smaller  scale,  viz.,  protuberances,  hollows  and  vortices.  The 
great  eruption  of  Krakatoa,  of  which  we  shall  have  more  to  say  later  on, 
probably  threw  up  a  column  of  air  directly  over  the  volcano  far  beyond 
the  limits  of  the  atmosphere,  most  of  which  was  lost.  Laplace,  in  the 
Mecanique  Celeste,  has  shown  from  certain  considerations  where  the  air  is 
supposed  to  extend  to  very  great  distances  from  the  earth,  that  the  ratio  of 
the  polar  to  the  equatorial  diameter  of  the  upper  surface  cannot  be  less  than 
two-thirds.  As  a  matter  of  fact,  since  the  atmosphere  does  not  extend  to 
any  great  distance,  the  ratio  is  much  greater  than  this,  not  differing  greatly 
from  that  of  the  earth. 

In  determining  the  height  of  the  atmosphere  we  shall  neglect  the  aqueous 
vapor.  A  cubic  meter  of  dry  air  at  o°  C.  and  a  pressure  of  760  mms.  of 
mercury,  that  is  under  standard  conditions,  weighs  1293.233  grammes  at 
Paris  (Regnault's  determination).  A  cubic  meter  of  aqueous  vapor  under 
the  same  conditions  weighs  five-eighths  as  much.  No  clouds  have  ever  been 


14  THE  ATMOSPHERE 

seen  above  seven  miles  from  the  earth's  surface  and  above  twelve  miles 
practically  no  vapor  exists.  Since  the  total  amount  of  vapor  in  the  atmos- 
phere is  but  a  small  fraction  of  its  total  mass  and  this  small  fraction  lies 
nearly  all  close  to  the  earth's  surface,  we  shall  make  no  very  appreciable  error 
in  taking  the  atmosphere  as  practically  dry. 

Since  the  total  height  of  the  atmosphere  is  small  compared  with  the  earth's 
radius,  we  shall  suppose  gravity  to  be  constant  throughout,  thereby  intro- 
ducing only  a  very  small  error.  We  shall  further  suppose  that  within  the 
limits  of  the  atmosphere  we  can  neglect  the  curvature  of  the  earth.  Our  data 
are  then  as  follows. 

At  the  bottom  of  the  atmosphere,  under  standard  conditions,  a  cubic 
meter  of  air  weighs  1293.233  grammes.  At  the  top  the  pressure  and  tem- 
perature are  zero  and  a  cubic  metre  of  air  weighs  ^  X  io~8  grammes.  Let 
us  consider  a  parallelopipedon  of  air  resting  upon  a  base  of  one  square  meter 
and  extending  vertically  from  the  surface  of  the  earth  to  the  superior  limit. 
If  we  suppose  the  temperature  to  be  constant  throughout  the  pressure  at  any 
level  will  be  equal  to  the  weight  of  the  superincumbent  mass  and  the  density 
of  the  air  at  this  level  will  be  proportional  to  the  pressure.  In  descending 
from  one  level  to  another  the  increment  of  pressure  will  be  equal  to  the 
weight  of  the  section  between  the  two  levels  and  is  proportional  to  the  height 

of  the  section  and  its  density.    Hence  A  p  =  -j>p.  A  h,  where  p  is  the  pres- 

sure, h  the  height  of  the  section,  and  K  is  some  constant  to  be  determined 
from  the  data.  Since  these  values  change  continuously,  we  may  write 

dp  =  jfp.  dh  or  dh  —  K  -£-.    Integrating  we  have  h  —  K  log.  ^p  where  p0 

•**•  P  Px 

is  the  pressure  at  the  lower  level,  px  that  of  the  upper  level.  Since  the  den- 
sity is  proportional  to  the  pressure  when  the  temperature  remains  constant, 

we  may  also  write  h  =  K  log.  -j=f.  If  we  take  the  height  of  our  section  as 

•LSf 

one  meter,  then  K  =  -  —  ,  from  which  we  can  determine  the  value  of  the 

fc** 

constant  K.  If  the  weight  of  a  cubic  meter  of  dry  air  at  the  given  tempera- 
ture and  pressure  is  w,  we  have  K  =  -  -  —  ;  —  . 


A 

At  the  surface  of  the  earth,  under  normal  conditions,  the  pressure  is 
10,332,790  grammes  on  every  square  meter  and  the  weight  of  a  cubic  meter 

of   air   is    1293.233    grammes.       Consequently   K  =  — 


log. 


=  18,409. 


103327907 


.00054318 


HEIGHT  15 

K  is  called  the  barometrical  coefficient.  It  is  constant  only  provided  the 
temperature  remains  uniform.  In  fact  K  is  a  function  of  the  temperature, 
and  is  nearly  proportional  to  the  absolute  temperature. 


P  273 

Since  w  =  1293.233  x  - — — X  — -•  where  r  is  the  absolute  tem- 

10332790        r 

perature, 


10332790 


r)    **•( 


X 


log.  (i  +  )•     From   this   formula    we  can  calculate 

the  value  of  K  for  any  temperature. 


90  100  10  20  30  40  50  60  70  80  00  200  10  20 


FIG.  2 


From  Fig.  2  we  see  that  K  is  practically  proportional  to  the  tem- 
perature, the  curve  being  nearly  a  straight  line.  The  accompanying  table 
gives  a  few  values  of  K  for  various  absolute  temperatures. 

We  shall  now  attempt  the  determination  of  the  height  of  the  atmos- 
phere at  the  equator  and  at  the  pole  under  average  conditions,  on  the  sup- 
position that  it  is  at  rest  relatively  to  the  earth.  The  average  temperature  at 


16  THE  ATMOSPHERE 

K  the   equator  is   27°   C,  and  the  average  pressure  758 

mms.    The  weight  of  a  cubic  meter  of  dry  air  under  these 
conditions  is 


3952 

5586  _  i  758       32.088 


.  00367x27 


59 

83 
103 

212 
247 
264 

273 
278 
291 
300 

pressure    per    square    meter    at    the    equator    is     10,305,925    grammes. 
Whence 


^944 

16666 

17821  since    32.088    is   the    average    value    of    gravity   at   the 

equator     (British    units),    and    32.1747    its    value    at 
Paris,   where   the   standard    determinations   were   made 


i8868 
19608 


20000  by    Regnault.      This    is    nearly    1175    grammes.      The 


v  T 

K  =  ; r-  =  =  20OOO. 

/      IJ75  \   .00005 

log.  I  i  H 

V    103059257 

If  the  temperature  remained  constant  throughout  we  should  have 

D  1 1 7  5 

h  —  K  log.  ~  =  20000  log. '-2-^  =  20000  log.  3X  io8X  1175  =  230,943 

Ux  -j  X   10 

meters,  or  about  one  hundred  and  forty-three  miles.  This  is  certainly  -very 
much  too  great,  since  the  temperature  decreases  to  absolute  zero  at  the 
upper  surface.  As  a  first  approximation,  therefore,  let  us  consider  the 
average  temperature  as  the  mean  of  the  extremes,  although  this  is  too  great. 
This  would  give  us  as  the  average  value  of  K,  10,000,'  and  the  total  height 
would  be  about  seventy-two  miles. 

If  we  start  from  the  pole,  we  have  as  the  average  initial  temperature 
— 18°  C.  with  perhaps  the  standard  pressure.  The  weight  of  a  cubic  meter 
of  air  under  these  conditions  is  1410  grammes  and  K  =  16,835.  On  the 
supposition  that  the  temperature  remains  constant,  h  =  195,729  meters,  or 
about  one  hundred  and  twenty-one  miles.  If  we  consider  the  average  tem- 
perature as  the  mean  of  the  extremes,  we  have  sixty  miles  as  the  height  of 
the  atmosphere  at  the  pole.  We  have  then  as  a  first  approximation,  seventy- 
two  miles  and  sixty  miles,  for  the  equator  and  pole  respectively,  as  limits 
within  which  the  atmosphere  must  lie. 

Our  knowledge  of  the  heights  of  average  isothermal  surfaces  is  very 
limited  and  derived  from  a  few  soundings  by  registering  balloons.  Ferrel 
some  years  ago  gave  the  annexed  table,  which  is  perhaps  a  rough  approxi- 
mation. 


HEIGHT  17 

TABLE  OF  ALTITUDES,  PRESSURES  AND  TEMPERATURE  BY  FERREL 


Temperature 
Centigrade 

Altitude 

Pressures 
Mms.  of  Mercury 

Kilometers 

Miles 

-f-2O 

0 

O 

760 

+  10 

2-5 

1.56 

565 

O 

5-0 

3-11 

416 

—  IO 

7-5 

4.66 

301 

20 

IO.O 

6.21 

217 

30 

12.5 

7-77 

153 

40 

15.0 

9-32 

108 

50 

17-5 

10.87 

75 

60 

20.  o 

12.42 

5i 

80 

25.0 

15-53 

27 

IOO 

30.0 

18.63 

9 

1  20 

35-o 

21-74 

3 

—  140 

40.0 

24.85 

i 

The  following  table  accords  more  with  the  latest  obtainable  data  and  is 
calculated  for  the  equator. 


Temperatures 

Heights 

Densities. 

Grammes 

jr 

per 

t\. 

Centigrade 

Absolute 

Meters 

Miles 

Cu.  Meter 

27 

300 

0 

o 

1209.6 

19608 

IO 

283 

2415 

4 

910.9 

18868 

0 

273 

4933 

3i 

669  .9 

17821 

-  18 

255 

7244 

4i 

497 

16666 

-  34 

239 

11268 

7 

285 

"5*79 

—  62 

211 

16097 

IO 

137 

In  trying  to  extend  this  table  beyond  the  observed  data,  we  have  at  hand 
two  methods.  We  can  draw  a  curve  connecting  the  heights  and  temperatures 
and  then  extend  it  in  the  manner  the  curve  tends  to  run — in  other  words,  we 
can  use  the  method  of  extrapolation — or  we  can  attempt  to  find  some  for- 
mula which  connects  the  known  values  and  then  predict  by  this  means  the 
unknown  values. 


i8 


THE  ATMOSPHERE 


Now  if  we  start  from  the  upper  surface  and  call  /  the  depth  of  any 
point  below  this  surface,  a  formula  seems  to  connect  the  depth  with  the 
temperature  quite  closely.  It  is  this,  t  =  Cl2,  where  T  is  the  absolute  tem- 
perature and  C  some  constant  to  be  determined  from  the  data.  Measuring 
our  depth  by  miles,  we  have  from  the  preceding  table  300  =  Cl2  and  211  = 
<T(/-io)2,  whence  C  =  .0732  and  /  =  64  miles.  This  is  the  total  depth  of  the 
atmosphere  at  the  equator. 


i  \'0~  i — i — i — i — i — r— i — i — i — r — r 

600  203  283  273  268  263  213  233  223  218  203  193 


173  163  153  143  138  123  113  103 
TEMPERATURES  ABSOLUTE 


73    63     63     43     83    23     13 


FIG.  3 


By  using  the  formula  r  =  .0732  I2,  we  have  for  the  absolute  tempera- 
ture 211°,  a  depth  of  54  miles ;  for  the  temperature  239°,  a  depth  of  57  miles ; 
for  the  temperature  255°,  a  depth  of  59  miles;  for  the  temperature  273°,  a 
depth  of  6 1  miles;  for  the  temperature  283°,  a  depth  of  62.2  miles;  and  for 
•the  temperature  300°,  a  depth  of  64  miles. 

The  formula  agrees  fairly  well  with  the  observed  facts  and  is  perhaps 
more  accurate  than  the  latter.  We  can  now  draw  the  curve  and  exemplify 


HEIGHT  19 

the  relation  graphically.    The  curve  is  a  parabola  with  its  vertex  at  the  upper 
limit. 

The  height  of  the  atmosphere  according  to  the  temperature  formula 
agrees  very  well  with  our  previous  determination.  Owing  to  centrifugal 
force  and  because  of  the  lesser  temperature  at  the  pole,  by  the  most  probable 
estimate  that  we  can  make,  the  height  of  the  atmosphere  at  the  equator 
should  be  about  fifteen  miles  greater  than  at  the  pole. 

We  shall  assume,  therefore,  as  the  most  probable  average  heights,  sixty- 
seven  miles  at  the  equator  and  fifty-two  miles  at  the  pole.  The  upper  sur- 

28 
face,  therefore,  would  be  a  spheroid  with  an  ellipticity  of  about ,  while 

the  ellipticity  of  the  lower  surface  is  that  of  the  earth,  viz., . 

We  have  arrived  at  the  above  estimate  on  the  assumption  that  the  atmos- 
phere cannot  become  less  dense  than  the  ether,  and  from  considerations 
already  given  we  have  calculated  that  the  density  of  the  ether  is  somewhat 
less  than  £  X  io~14  that  of  water.  If  we  apply  in  our  calculation  the  esti- 
mate of  Graetz,  viz.,  that  the  ether  is  io-17  the  density  of  water,  we  shall 
obtain  eighty  miles  as  the  average  extension  of  the  atmosphere. 

While  it  will  always  be  impossible  for  us  to  measure  directly  the  limit 
of  the  atmosphere,  there  are,  however,  phenomena  through  which  indirectly 
it  has  been  sought  to  obtain  an  estimate.  These  are  shooting  stars,  auroras, 
the  duration  of  twilight,  and  phosphorescent  phenomena  in  the  upper  atmos- 
phere. The  following  quotations  will  serve  to  give  an  idea  as  to  what 
deductions  have  been  drawn  from  these  phenonema. 

"Meteorites  have  been  seen  at  a  height  of  two  hundred  miles,  and  as 
their  luminosity  is  undoubtedly  due  to  friction  against  the  air,  there  must 
be  air  at  such  a  height.  This  higher  estimate  is  supported  by  observations 
made  at  Rio  Janeiro  on  the  twilight  arcs  by  M.  Liais,  who  estimated  the 
height  of  the  atmosphere  at  between  one  hundred  and  ninety-eight  and  two 
hundred  and  twelve  miles.  The  question  as  to  the  exact  height  of  the  atmos- 
phere must,  therefore,  be  considered  as  awaiting  settlement." — Ganot's 
"Physics." 

"But  observations  on  meteors  show  that  the  atmosphere  really  extends 
to  a  height  of  at  least  one  hundred  miles,  and  indeed  at  that  height  is  suffi- 
ciently dense  to  cause  rapid  combustion  of  a  meteor  passing  through  it. 
Observations  on  the  aurora  lead  at  least  to  a  suspicion  that  this  phenomenon 
sometimes  takes  place  at  a  height  of  three  hundred,  four  hundred  or  five 
hundred  miles.  We  could  hardly  suppose  it  to  occur  in  an  absolute  vacuum, 
though  it  would  be  unsafe  to  infer  from  this  that  the  medium  in  which  it 
occurs  is  an  extension  of  the  atmosphere  proper." — Professor  Newcomb  in 
Universal  Cyclopaedia. 

"The  height  of  the  aurora  above  the  surface  of  the  ground  is  probably 


2O  THE  ATMOSPHERE 

lower  than  has  been  generally  stated.  Lemstrom  holds  that  from  twenty-two 
to  forty-four  miles  is  a  close  approximation  to  the  truth ;  and  it  may  be 
regarded  as  certain  that  even  in  more  southern  latitudes  the  aurora  is  often 
seen  much  lower — at  a  height  of  two  or  three  miles,  for  instance.  In  polar 
countries  certain  forms  of  aurora,  more  especially  those  of  weak  flames,  are 
seen  to  proceed  from  the  ground  on  the  tops  of  certain  mountains." — Ganot's 
"Physics." 

"A  meteor  seen  over  New  England  on  September  6,  1886,  and  reported 
by  several  observers  of  the  New  England  Meteorological  Society,  was  de- 
termined by  Professor  Newton  of  Yale  College  to  have  become  visible  at 
an  altitude  of  ninety  miles  over  northwestern  Vermont  and  to  have  disap- 
peared at  an  altitude  of  twenty-five  miles  over  southeastern  New  Hamp- 
shire."— W.  M.  Davis  in  "Elementary  Meteorology." 

"The  method  of  twilight  arcs  to  determine  the  height  of  the  atmosphere, 
first  devised  by  Kepler,  gives  our  atmosphere  a  height  of  from  thirty  to 
thirty-seven  miles." — Flammarion  in  "The  Atmosphere." 

"It  is  to  be  noted  that  different  methods  give  different  heights  for  the 
atmosphere,  but  there  is  no  discrepancy,  as  different  things  are  meant.  Thus, 
if  experiments  on  twilight  give  forty  miles  as  the  height,  this  implies  that  the 
air  above  this  elevation  reflects  no  appreciable  amount  of  light ;  while  if  we 
define  the  height  to  be  the  point  where  the  friction  will  not  set  light  to  a 
meteor,  we  have  about  seventy  miles ;  but,  of  course,  there  is  no  reason  why 
there  should  not  be  some  air  at  much  greater  heights." — Glaisher. 

"In  the  year  1798  an  investigation  of  the  heights  of  shooting  stars  was 
undertaken  by  Brandes  at  Leipzig  and  by  Benzenberg  at  Diisseldorf.  Having 
selected  a  base  line  (about  nine  miles  in  length),  they  placed  themselves  at  its 
extremities  on  appointed  nights  and  observed  all  the  shooting  stars  which 
appeared,  tracing  their  courses  through  the  heavens  on  a  celestial  map,  and 
noting  the  instants  of  their  appearances  and  extinctions  by  chronometers  pre- 
viously compared.  Similar  sets  of  observations  which  have  since  been  re- 
peated lead  to  the  conclusion  that  the  heights  of  the  shooting  stars  above  the 
ground  vary  from  six  to  five  hundred  and  fifty  English  miles,  that  they  move 
with  a  velocity  of  between  eighteen  and  two  hundred  and  twenty  miles  a 
second,  and  that  their  trajections  are  frequently  not  straight  lines.  It  may 
be  asked,  how  the  evolution  of  light  is  possible  at  altitudes  so  far  surpassing 
the  probable  bounds  of  the  atmosphere,  or  where,  supposing  air  to  exist,  it 
must  necessarily  be  so  attenuated  as  to  approach  the  limits  of  absolute 
vacuity  ?  Poisson,  the  eminent  French  geometer,  has  endeavored  to  solve  the 
question  by  affirming  the  probability  of  an  atmosphere  of  electricity  sur- 
rounding the  earth  and  lying  above  the  atmospheric  air." — Hartwig,  1893, 
"The  Aerial  World." 

"After  the  great  eruption  of  Krakatoa,  in  1883,  the  brilliant  sunset  glows 
and  the  longer  twilight  showed  that  the  dust  emitted  by  the  eruption  remained 


HEIGHT  21 

for  more  than  a  year  suspended  at  a  height  of  at  least  sixty  miles.  The 
so-called  "luminous  clouds"  seen  at  night  during  the  same  period,  and  which 
was  probably  the  same  dust  still  illumined  by  the  sun,*  were  found  by 
trigonometrical  measurements  to  have  about  the  same  altitude.  Although  it 
is  computed  that  at  a  height  of  seventy  miles  the  air  has  less  than  one- 
millionth  of  its  density  at  sea  level,  it  is  there  sufficiently  dense  to  render 
meteors  luminous  by  friction." — A.  L.  Rotch,  "Sounding  the  Ocean  of  Air." 

"The  height  of  these  meteors  has  been  found  from  simultaneous  trigo- 
nometrical measurements  sometimes  to  exceed  one  hundred  miles." — Rotch. 

"The  gases  composing  the  atmosphere  probably  extend  to  heights  pro- 
portionalf  to  their  density,  viz.,  oxygen  to  thirty  miles,  and  nitrogen  to 
thirty-five  miles,  although  water  vapor  nearly  disappears  at  twelve  miles." — 
Rotch. 

"From  these  considerations  it  is  supposed  that  the  atmosphere  vanishes, 
as  measured  by  the  barometer,  at  about  thirty-eight  miles,  and  this  is  about 
the  height  indicated  by  twilight,  which  is  the  reflected  light  of  the  sun  when 
18°  below  the  horizon." — Rotch. 

We  might  multiply  these  quotations  indefinitely,  but  the  above  are  suffi- 
cient. The  results  are  extremely  discordant,  often  very  improbable,  and  at 
times  impossible.  Thus  the  finding  of  luminous  meteors  at  a  height  of  five 
hundred  and  fifty  miles  moving  with  a  velocity  of  two  hundred  and  twenty 
miles  a  second  is  clearly  impossible.  For  astronomical  reasons  the  motion 
of  a  meteor  relatively  to  the  earth  cannot  exceed  forty  miles  a  second.  A 
little  consideration  will  show  that  it  is  impossible  for  two  distant  observers 
to  trace  accurately  the  path  of  a  meteor  against  the  stars  during  its  momen- 
tary flight.  The  hosts  of  small  flashes  during  a  clear  night  would  make  it 
almost  impossible  for  them  to  identify  the  same  object,  and  communication 
(telegraphic  or  telephonic)  would  be  of  no  assistance.  At  a  hundred  miles 
above  the  earth,  if  there  were  any  air,  its  density  would  certainly  be  less 
than  that  of  the  ether,  and  hence  we  may  be  satisfied  that  no  meteor  has  ever 
been  seen  at  this  height.  The  method  is  not  one  susceptible  of  accuracy,  and 
hence  without  any  hesitation  we  may  throw  aside  all  computations  deduced 
from  the  flight  of  meteors. 

The  method  of  twilight  arcs  is  even  in  theory  incapable  of  indicating  the 
upper  surface  of  the  atmosphere,  for  these  layers  reflect  no  light.  We  can 
get  no  reflection  of  the  sunlight  from  the  air  in  a  room  unless  it  contains 
foreign  particles  (motes).  The  principle  involved  can  be  seen  from  Fig.  4. 

The  horizon  at  the  point  A  is  ADB.  When  the  sun  gets  below  the 
horizon  some  light  will  still  be  reflected  to  A  from  the  upper  layers.  If  the 


*More  probably  phosphorescence. — Author. 

t  Probably  "inversely  proportional"  is  meant. — Author. 


22 


THE  ATMOSPHERE 


point  D  is  in  the  highest  layer  capable  of  reflecting,  then  the  line  DC,  in 
which  rays  from  the  sun  illumine  the  point  D,  and  are  tangent  to  the  earth, 
is  the  limit  of  reflection.  Beyond  this  point  no  light  can  be  reflected  directly 
to  A.  The  angle  BDC  is  called  the  twilight  arc.  Refraction  is  neglected,  the 


effect  of  which  would  be  to  increase  greatly  the  twilight  arc.  The  twilight 
arc  is  usually  about  18°,  but  it  varies  with  time  and  place  and  with  the 
observer.  Of  course,  it  is  not  possible  to  set  a  sharp  limit  to  it,  since  it  fades 
away  gradually,  and  long  after  the  glow  has  ceased  to  be  perceived,  light  is 
still  being  reflected.  The  angle  ODA  is  thus  81°,  and  calling  h  the  height  of 
D  above  the  earth,  since  the  earth's  radius  is  four  thousand  miles,  we  have 

R  4000 

-75— — j- =  —  — 7  =  sin  8 1  =  .98769.  From  which  we  see  that  h  =  50. 
R  +  k  4000  +  k 

From  such  a  calculation  it  has  been  deduced  that  the  height  of  the  atmos- 
phere is  fifty  miles.  But  the  effect  of  refraction,  as  can  easily  be  seen,  is 
to  bring  down  the  highest  point  from  which  reflections  are  perceived  by  a 
considerable  amount.  Furthermore,  the  air  itself  is  phosphorescent.  That 
is  to  say,  that  after  absorbing  the  strong  radiation  of  the  sun  all  day  it  gives 
out  radiation  again  during  the  night,  both  luminous  and  non-luminous, 
though  with  decreasing  intensity.  It  is  for  this  reason  that  we  are  able  to 
find  our  way  along  roads  in  the  darkest  nights  even  when  there  are  no  stars. 
It  is  not  absolutely  dark,  owing  to  this  weak  diffused  illumination  from  the 
heavens,  but  the  darkness  increases  with  the  lapse  of  time  after  the  with- 
drawal of  the  sun.  The  old  folk  saying  "It  is  darkest  before  the  dawn"  has 
something  of  a  scientific  foundation.  The  twilight  glow,  therefore,  fades 
gradually  into  this  general  diffuse  illumination  of  the  heavens.  From  the 


HEIGHT  23. 

above  discussion  it  will  be  seen  that  the  method  of  the  twilight  arc  is  totally 
imsuited  to  the  determination  of  the  height  of  the  atmosphere. 

The  phenomenon  of  the  aurora,  if  not  identical  with,  is  closely  analogous 
to  that  of  the  discharge  of  electricity  through  gases.  As  such  it  must  be 
strictly  limited  to  our  atmosphere.  In  the  laboratory  it  is  possible  to  imitate 
this  glow  in  an  exhausted  tube,  but  when  the  exhaustion  is  carried  below  a 
certain  point  all  discharge  ceases.  This  limiting  density  is  much  above  that 
of  the  ether.  According  to  Lemstrom,  this  glow  is  never  seen  at  a  greater 
height  than  forty-four  miles,  and  in  all  probability  this  is  not  far  from  the 
truth. 

We  have  seen  that  neither  by  the  phenomenon  of  the  aurora,  nor  of 
meteors,  nor  yet  of  the  twilight  arc,  have  we  a  suitable  means  of  determining 
the  height  of  the  atmosphere.  There  yet  remains  the  phenomenon  of  the 
luminous  clouds  which  were  seen  for  a  long  time  after  the  eruption  of 
Krakatoa  and  which  we  have  already  mentioned.  As  a  result  of  the  ex- 
plosion the  dust  and  gases  from  the  volcano  were  blown  clear  through  the 
atmosphere  and  a  part  of  them  lost  in  space.  There  were  thus  extraneous 
particles  (dust  and  frozen  gases)  introduced  into  the  higher  layers  clear  up 
to  the  limiting  surface.  These  particles  maintained  their  levels  for  a  long" 
time,  sinking  very  slowly.  Trigonometrical  measurements  were  taken  of 
these  clouds,  with  sufficient  bases  and  under  conditions  favoring  extreme 
accuracy.  They  were  found  to  be  at  least  sixty  miles  above  the  earth,  and 
the  limit  of  luminosity  probably  coincided  very  nearly  with  the  limit  of  the 
atmosphere.  This  is  practically  identical  with  our  estimate  of  sixty-seven 
miles.  That  is  to  say,  we  have  positive  evidence  that  the  atmosphere  extends 
to  at  least  sixty  miles  at  the  equator,  and  as  shortly  beyond  this  height  it 
acquires  a  density  equal  to  that  of  the  ether  and  the  temperature  of  space,  we 
shall  in  all  probability  be  quite  close  to  the  truth  in  assigning  it  such  a  height. 
The  most  probable  height,  therefore,  seems  to  be  about  sixty-seven  miles  at 
the  equator,  and  fifty-two  miles  at  the  pole,  and  we  shall  assume  these  values 
as  the  height  of  the  atmosphere. 

CONVECTIVE  EQUILIBRIUM 

We  shall  next  take  up  the  subject  of  convective  equilibrium.  We  can 
suppose  an  ideal  case  in  which  the  air  is  heated  by  conduction  from  the 
ground  only,  and  ris'es  through  the  cold  upper  strata.  It  afterwards  neither 
receives  nor  loses  any  heat,  either  by  radiation  or  conduction.  As  it  rises  the 
pressure  decreases  and  consequently  it  expands  and  becomes  cooler.  After 
the  process  has  gone  on  for  a  sufficient  time  a  condition  of  convective  equi- 
librium will  be  established,  where  if  we  move  a  portion  of  the  air  from  one 
point  to  any  other,  it  will  remain  indifferently  in  equilibrium,  having  no 
tendency  to  change  its  level. 


24  THE  ATMOSPHERE 

Since  dp  =  —  dk.  D,  where  D  is  the  density,  and  by  the  adiabatic 

Dk 
law  P  =  po  -ry-jt,  where  p0  and  D0  are  the  pressure  and  density  at  the  sur- 

~*9 

face  of  the  earth,  and  k  is  the  ratio  of  the  two  specific  heats,  we  have 
~l  dD  =  -dh.D. 


Whence,  h  =  „  /°k — r  (Dk~l  -  D" 

D*  (k-  i)  \ 

Since  under  normal  conditions  p0=  10,332,790  grammes   per   square 
meter,  and  the  density  of  a  cubic  meter  of  air  is  1293  grammes  approximately, 

p   k 
we  find  that  ^  &/!  _ — 7  =  1 456.451,  the  value  of  k  being  1.41.     Since  the 

density  becomes  zero  at  the  upper  limit,  the  total  height  will  be  1456.451  X 
1293 -41  =  27,482  meters,  or  seventeen  miles,  which  we  know  to  be  far  from 
the  truth.  Since  by  the  thermodynamic  law  for  adiabatic  changes, 

ID  \k-1 

T  =  TO  I  -~- 1      ,  and  since  the  depth  or  distance  from  the  upper  surface  is 


-Dk'',  it  follows  that  the  absolute  temperature  is  proportional  to 


n  >*(b        \ 

the  depth.  Hence,  under  this  ideal  condition,  which  is  represented  by  the 
formula  h  =  C  (D*-*  —  Dk~l)  the  barometric  coefficient  C  remains  constant 
throughout  and  the  temperature  is  proportional  to  the  depth.  The  density 
and  the  temperature  are  both  zero  at  the  upper  limit.  If  under  such  condi- 
tions a  mass  of  air  were  introduced  of  a  higher  temperature  than  the  level  at 
which  it  was  placed,  it  would  continue  to  rise  to  the  top,  and  conversely  any 
air  colder  than  its  level  would  sink  to  the  bottom. 

Of  course,  such  an  adiabatic  condition  could  not  exist  in  the  atmosphere, 
since  the  air  gains  and  loses  heat  at  all  levels,  both  by  radiation  and  conduction. 
The  atmosphere  at  all  levels  is  considerably  warmer  than  would  be  the  case 
under  adiabatic  conditions  ;  and  consequently  a  mass  of  air  warmer  than  its 
surroundings  quickly  reaches  a  level  where  the  temperature  and  pressure, 
and  hence  the  density,  are  the  same.  As  during  this  short  assent  not  much 
heat  is  lost  either  by  conduction  or  radiation,  the  mass  approximates  some- 
what to  an  adiabatic  rate  of  cooling. 

We  have  seen  that  the  atmosphere  lies  in  a  comparatively  thin  sheet  over 
the  earth,  its  average  height  being  about  sixty  miles.  The  level  of  half  mass 
is  at  17,400  ft.,  or  half  the  total  mass  of  the  atmosphere  lies  within  this  is- 
tance  of  the  earth. 

VARIOUS    ATMOSPHERES 

We  have  seen  that  the  height  of  the  atmosphere  may  be  derived  to  a 

fC  T~) 

first  approximation  from  the  formula  H  '=  —  -  log.  —  ,  where  K0,   D0  are 

2  */ 


HEIGHT 


the  barometric  coefficient  and  density  at  the  bottom,   and   Du  is   the 
density  at  the  top. 

Or,  H  =  .,  3  -  -ft,  where  w  is  the  weight  of  unit  volume  at 

* 


the  bottom,  and  W  is  the  weight  of  the  total  mass  of  the  atmosphere. 

Since  w,  the  density  at  the  bottom,  is  proportional  to  W,  the  denomina- 
tor of  the  expression  above  does  not  change  value  with  the  mass  of  a  gaseous 
envelope  surrounding  an  attracting  body.  In  other  words,  if  we  poured  a 
certain  mass  of  a  simple  gas  around  an  attracting  body,  or  twice  the  mass 
or  ten  times  the  mass,  the  barometric  coefficient  would  in  all  cases  remain 
the  same.  But  w,  where  W  remains  the  same,  would  change  for  different 
simple  gases,  since  it  is  the  specific  density.  It  is  evident  then  from  the 
expression  above  that  the  height  of  the  atmosphere  due  to  a  simple  gas 
increases  as  the  mass  of  the  gas  increases,  but  it  increases  very  slowly  with 
the  mass.  On  the  other  hand,  for  the  same  mass  of  two  different  gases,  the 
atmosphere  of  the  less  dense  gas  will  be  higher  than  that  of  the  other,  and 
the  difference  of  the  heights  will  increase  rapidly  with  the  difference  of  the 
densities.  The  barometric  coefficient  for  oxygen  at  o°  C.  is  16,666,  and  for 
nitrogen  at  the  same  temperature  18,518.  An  atmosphere  of  oxygen  of  the 
same  mass  as  the  atmosphere,  and  having  a  uniform  temperature  of  o°  C.  at 
the  surface  of  the  earth,  would  rise  to  a  height  of  sixty  miles,  while  an 
atmosphere  of  oxygen  consisting  of  only  one-fifth  of  this  mass,  would,  under 
the  same  conditions,  rise  to  a  height  -of  56^  miles.  An  atmosphere  of 
nitrogen  of  the  same  mass  as  the  air  and  at  o°  C.  at  the  bottom,  would 
extend  to  sixty-seven  miles,  while  four-fifths  of  this  mass  would  extend  to 
sixty-six  miles.  Now,  according  to  Dalton's  law,  each  gas  in  the  atmos- 
phere forms  a  separate  atmosphere  just  as  if  it  existed  alone.  In  forming 
a  mixed  atmosphere,  therefore,  by  taking  masses  of  nitrogen  and  oxygen  in 
the  ratio  of  4  to  I,  as.  is  the  case  with  air,  we  should  have  an  atmosphere  of 
nitrogen  extending  sixty-six  miles,  and  an  independent  atmosphere  of 
oxygen  extending  56^  miles.  Consequently,  there  would  be  practically  no 
oxygen  in  the  upper  limits,  and  the  proportion  of  oxygen  to  nitrogen  would 
decrease  slightly  with  the  height.  Further,  the  dissipation  of  the  atmos- 
phere at  the  upper  level  is  practically  at  the  expense  of  the  nitrogen,  and  the 
atmosphere  becomes  proportionally  richer  in  oxygen  continually. 

Measurements  seem  to  show  that  there  is  a  decrease  in  the  proportion 
of  oxygen  with  the  height,  though  within  a  few  miles  of  the  earth  this  is 
scarcely  perceptible. 

AQUEOUS    VAPOR 

The  aqueous  vapor  in  the  atmosphere  cannot  in  any  sense  be  considered 
as  a  separate  atmosphere.  Assuming  the  law  r  =  C/2,  we  find  that  the  tern- 


26 


THE  ATMOSPHERE 


perature  of  the  atmosphere  falls  much  more  rapidly  than  the  pressure. 
Without  entering  into  the  calculation,  we  may  say  that,  near  the  surface  of 

578 


the  earth,  for  every  meter  that  we  ascend  the  temperature  falls 

T    \ 

degree  Centigrade,  while  the  fall  in  pressure  is  only 


n*,6S3° 

This  is  a  much 


r       i        i        i 

01  o 

PRESSURES  IN  MM.  OF  MERCURY 
FIG.  5 

more  rapid  rate  than  the  fall  of  temperature  for  maximum  pressure  of  vapor 
at  ordinary  temperatures. 

In  the  accompanying  Fig.  5  we  have  represented  the  average  fall  of 
pressure  with  the  temperature,  starting  from  different  pressures  of  the  mag- 
nitude of  vapor  pressures,  as  it  occurs  in  the  atmosphere.  The  vapor  curve, 
sometimes  called  the  "steam  line,"  shows  the  maximum  tension  of  vapor 


HEIGHT 


27 


possible  for  the  corresponding  temperatures.     It  is,  in  fact,  the  saturation 
curve. 

The  base  line  represents  a  temperature  of  16°  C.,  the  average  tempera- 
ture of  the  earth's  surface.  The  vapor*  curve  practically  meets  the  axis  of 
temperature  at  — 80°  C.,  although  theoretically  some  vapor  exists  even  at 
— 273°.  If,  now,  we  carry  a  mass  of  vapor  from  the  base  temperature, 
16°  C.,  along  one  of  the  pressure  curves,  it  will  be  subsaturated  while  it  is 
within  the  vapor  curve.  On  crossing  the  vapor  curve  it  is  exactly  saturated, 
while  above  the  vapor  curve  condensation  results.  Consequently  a  mass  of 
vapor,  endeavoring  to  arrange  itself  as  an  atmosphere  about  the  earth,  would 
be  successful  until  it  met  the  vapor  curve,  when  there  would  be  a  sharp 
discontinuity.  If  we  started  with  the  maximum  tension,  the  discontinuity 
would  occur  at  once.  At  a  height  of  ten  miles  not  more  than  one-fifth 
gramme  per  cubic  meter  could  exist  as  vapor,  and  the  amount  decreases 
rapidly  with  the  height.  Theoretically,  a  trace  of  vapor  should  exist  eyen 
at  the  upper  limit,  but  it  is  impossible  to  detect  any  by  means  of  instruments 
at  a  comparatively  short  distance  above  the  earth. 


PRESSURES  IN  MM.  OF  MERCURY  AT  SATURATION  FOR  DIFFERENT 
TEMPERATURES   OF  AQUEOUS    VAPOR 


Temperature 

Force  of  Vapor, 

Temperature 

Force  of  Vapor, 

Centigrade 

Mms. 

Centigrade 

Mms. 

-32° 

0.32 

10° 

9.17 

—20° 

•93 

15° 

12.70 

—10° 

2.09 

20° 

17-39 

-5° 

3-n 

25° 

23-55 

0° 

4.60 

30° 

31-55 

+5° 

6-53 

40° 

54-91 

Glaisher  states  that  he  found1  the  relative  humidity  to  increase  slightly 
up  to  half  a  mile,  after  which  it  decreased,  while  at  five  miles  he  could 
detect  no  vapor.  Rotch  states  that  water  vapor  practically  disappears  at 
twelve  miles.  For  an  adiabatic  expansion  of  the  vapor,  the  heights  at  which 
it  could  exist  would  be  considerably  lessened. 

The  average  amount  of  vapor  in  the  atmosphere  has  not  been  deter- 
mined. While  enormous  in  the  aggregate,  still  compared  to  the  other  con- 
stituents it  is  not  much.  Practically  all  of  it  is  within  a  very  short  distance 

*  The  vapor  curve  is  very  closely  represented  by  the  empirical  formula,  due  to 
Rankine,  p  =  C  (&  —  233)*,  for  ordinary  temperatures.  For  very  low  temperatures  it 
does  not  apply. 


28  THE  ATMOSPHERE 

from  the  earth,  and  most  of  this  is  in  the  tropics  and  the  temperate  zones. 
There  is  comparatively  little  in  the  polar  tracts.  It  is  most  irregularly  dis- 
tributed, great  islands  and  continents  of  it  floating  about  in  some  regions, 
while  in  others  there  are  gaps  or  it  is  practically  wanting.  It  is  always 
undergoing  a  ceaseless  round  of  condensation  and  evaporation.  Owing  to 
the  slowness  of  the  process  of  diffusion,  a  mass  of  vapor  may  remain  isolated 
for  long  periods,  and  a  high  mountain  range  is  often  sufficient  to  fence  it 
off  effectually  from  vast  tracts. 

The  specific  gravity  of  vapor  relatively  to  air  is  five-eighths ;  conse- 
quently warm  moist  air,  which  is  capable  of  holding  more  vapor  than  colder 
air,  is  lighter  than  dry  air  and  has  its  ascensional  power  still  further 
increased  by  its  vaporous  constituent.  Vapor  is  preeminently  the  dynamical 
element  of  the  atmosphere.  From  its  instability  it  is  the  chief  factor  in 
furthering  the  more  violent  phenomena,  such  as  cyclones,  tornadoes,  thunder- 
storms, etc. 

'PRESSURE 

The  pressure  of  the  air  at  any  point  is  measured  by  the  barometer.  If 
the  atmosphere  were  perfectly  still  the  pressure  would  be  some  function  of 
the  depth  below  the  upper  level  surface.  But  since  the  atmosphere  is  in 
ceaseless  movement,  on  the  fundamental  statical  element  there  are  superposed 
transient  and  mostly  slight  dynamical  elements.  Thus,  strictly,  the  barometer 
is  not  an  indicator  of  the  height  or  density  of  the  atmosphere  overhead 
alone ;  it  measures  the  pressure  and  nothing  more.  If  we  expose  the  face  of 
an  aneroid  perpendicularly  to  a  violent  gust  of  wind,  it  immediately  shows 
an  increase  of  pressure,  and  if  we  turn  its  back  to  the  gust  there  will  be  a 
decrease  of  pressure,  while,  of  course,  the  total  height  and  density  of  the 
atmosphere  have  not  changed.  The  pressure  of  the  wind  against  a  squarely 
opposing  surface  is  roughly  proportional  to  the  surface  area  and  the  square 
of  the  velocity  of  the  wind.  We  can  see  why  this  should  be  so  since  the 
impact  of  the  wind  is  proportional  to  the  mass  times  the  velocity,  and  the 
mass  or  amount  of  air  striking  in  a  given  time  is  proportional  to  the  velocity ; 
hence,  the  total  force  is  proportional  to  the  square  of  the  velocity.  An 
ascending  current  of  air  will  naturally  decrease  the  pressure  below,  while  a 
descending  current  will  increase  it. 

At  the  equator  there  is  an  extremely  regular  daily  oscillation  of  the 
pressure.  It  is  a  maximum  at  the  equator,  both  for  amplitude  and  regu- 
larity, but  is  easily  discernible  through  the  torrid  zones,  becoming  finally 
insensible  in  the  temperate  zones.  At  10  A.M.  at  the  equator  the  barometer 
stands  highest,  at  4  P.M.  lowest.  Again  at  10  P.M.  there  is  another  maximum, 
while  at  4  A.M.  occurs  a  second  minimum,  the  amplitude  amounting  some- 
times to  three-twenthieths  of  an  inch.  So  regular  is  this  oscillation  in  the 
tropics  that  it  is  said  to  be  possible  to  tell  the  time  of  day  by  the  barometer 


HEIGHT  29 

within  a  quarter  of  an  hour.  Here  we  have  a  phenomenon  plainly  depending 
upon  the  sun.  The  maxima  and  the  minima  are  spaced  at  six  hour  inter- 
vals. The  low  barometers  occur  when  the  temperature  is  highest  for  the 
day  and  when  it  is  lowest.  The  high  barometers  occur  at  a  time  when  the 
temperature  is  the  mean  of  the  day.  The  explanation  is  as  follows.  In  the 
tropics  the  heating  effects  of  the  sun's  rays  is  chiefly  at  the  surface  of  the 
earth,  where  the  lower  layers  of  the  atmosphere  become  very  hot.  The 
upper  layers  are  in  fact  very  diathermanous  and  are  little  affected.  The 
first  effect  of  the  sun's  rays,  therefore,  is  an  expansion  of  the  lower  layers. 
This  expansion  is  opposed  by  the  inertia  of  the  upper  layers  and  until  the 
tension  is  relieved,  the  tension  mounts  steadily.  At  10  A.M.  there  is  an 
equilibrium  between  the  rate  of  expansion  and  the  flow  upward,  and  conse- 


FIG.  6 

quently  the  tension  remains  momentarily  stationary.  As  the  temperature 
increases  the  upward  flow  exceeds  the  expansion,  and  the  tension  falls.  This 
upward  movement  is  a  maximum  at  4  P.M.  when  the  temperature  is  greatest. 
Directly  after  this  the  lower  layers  begin  to  cool  and  the  upper  layers  to 
fall  back.  This  downward  movement  reaches  a  maximum  at  10  P.M.,  and 
at  that  time  the  tension  becomes  a  maximum  again.  After  this  there  is  a 
rebound  and  the  compressed  lower  layers  are  relieved  by  a  second  upward 
movement,  which  reaches  its  maximum  at  4  A.M.  Here  occurs  the  second 
minimum  at  the  coldest  part  of  the  day.  The  greater  maximum  occurs  at 
10  A.M.  and  the  greater  minimum  at  4  P.M.  These  are  the  chief  ones,  the 
secondary  ones  being,  as  it  were,  mere  echoes  of  the  first. 

The  oscillation  is  shown  graphically  by  the  accompanying  diagram. 
(Fig.  6.)  The  observations  were  taken  in  the  Pacific  in  Lat.  i°  S. 

The  case  is  somewhat  analogous  to  a  man  standing  on  the  platform  of 


3O  THE  ATMOSPHERE 

a  weighing  scales  and  jumping  upward.  As  he  jumps  the  beam  goes  up; 
while  he  is  in  the  air  the  beam  descends ;  when  he  falls  back  on  to  the  plat- 
form the  beam  goes  up  again;  after  which  the  beam  falls  again  and  the 
process  repeats  itself  anew.  The  air  springs  from  the  barometer,  sending 
it  up ;  after  the  air  has  acquired  its  upward  momentum  the  barometer  sinks ; 
falling  back  again  on  the  barometer  it  sends  it  up;  after  which  with  the 
cessation  of  the  downward  movement  it  falls  again. 

Dr.  Buchan  puts  it  thus : 

"Since  the  two  maxima  of  daily  pressure  occur  when  the  temperature 
is  about  the  mean  of  the  day,  and  the  two  minima  when  it  is  at  its  highest 
and  lowest  respectively,  there  is  suggested  a  connection  between  the  daily 
barometrical  oscillations  and  the  daily  march  of  temperature;  and  similarly 
a  connection  with  the  daily  march  of  the  amount  of  vapor  and  humidity  in 
the  air.  The  view  entertained  by  many  of  the  causes  of  the  daily  oscillations 
may  be  thus  stated :  the  forenoon  maximum  is  conceived  to  be  due  to  the 
rapidly  increasing  temperature,  and  the  rapid  evaporation  owing  to  the 
great  dryness  of  the  air  at  this  time  of  the  day,  and  to  the  increased 
elasticity  of  the  lowermost  stratum  of  air  which  results  therefrom,  until  a 
steady  ascending  current  has  set  in.  As  the  day  advances  the  vapor  becomes 
more  equally  diffused  upwards  through  the  air,  an  ascending  current,  more 
or  less  strong  and  steady,  is  set  in  motion,  a  diminution  of  elasticity  fol- 
lows, and  the  pressure  falls  to  the  afternoon  minimum. 

"From  this  point  the  temperature  declines,  a  system  of  descending  cur- 
rents sets  in,  and  the  air  of  the  lowermost  stratum  approaches  more  nearly 
the  point  of  saturation,  and  from  the  increased  elasticity,  the  pressure  rises 
to  the  evening  maximum.  As  the  deposition  of  dew  proceeds,  and  the  fall 
of  temperature  and  consequent  downward  movement  of  the  air  are  arrested, 
the  elasticity  is  again  diminished  and  the  pressure  falls  to  the  morning 
minimum." 

And  again :  "Some  time  elapses  before  the  higher  expansive  force  called 
into  play  by  the  increase  of  temperature  can  counteract  the  vertical  resist- 
ance it  meets  from  the  inertia  and  viscosity  of  the  air.  Till  this  resistance 
is  overcome,  the  barometer  continues  to  rise,  not  because  the  mass  of  atmos- 
phere overhead  is  increased,  but  because  a  higher  temperature  has  increased 
the  tension  or  pressure.  When  the  resistance  has  been  overcome,  an  ascend- 
ing current  of  the  warm  air  sets  in,  the  tension  begins  to  be  reduced  and  the 
barometer  falls  and  continues  to  fall  till  the  afternoon  minimum  is  reached. 
Thus  the  forenoon  maximum  and  the  afternoon  minimum  are  simply  a  tem- 
perature effect,  the  amplitude  of  oscillation  being  determined  by  latitude,  the 
quantity  of  aqueous  vapor  overhead,  and  the  sun's  place  in  the  sky." 

Lord  Kelvin*  has  suggested  that  the  daily  oscillation  may  be  considered 

*  Proceedings  of  the  Royal  Society  of  Edinburgh,  1882. 


HEIGHT  31 

as  a  tide  produced  by  the  increase  of  temperature  resulting  from  solar 
heat.  The  oscillation,  therefore,  would  be  a  forced  wave  completing  the 
circuit  of  the  earth  in  twenty-four  hours,  just  as  the  attractional  tide  of  the 
ocean  sweeps  around  in  nearly  that  period.  He  finds  that  the  free  wave 
in  the  atmosphere  would  have  a  period  something  like  thirteen  hours  or 
about  half  the  diurnal  period,  and  that  "the  free  oscillation  produced  by  a 
relatively  small  amount  of  tide-producing  force  will  have  an  amplitude  which 
is  larger  for  the  half-day  term  than  for  the  whole-day  term." 

But  the  barometric  oscillation  is  in  no  sense  a  wave  with  a  surface 
velocity,  and,  therefore,  not  a  tide  ;  it  is  simply  a  vertical  oscillation  with  no 
lateral  components.  If  we  supposed  the  earth  at  rest  and,  by  properly  screen- 
ing the  sun,  produced  the  same  temperatures  in  the  atmosphere  on  the  ex- 
posed side,  we  should  have  the  same  vertical  oscillations,  but  no  lateral  waves 
or  tides. 

PRACTICAL    BAROMETRY 

The  determination  of  a  height  by  the  barometer  is  equivalent  to  weigh- 
ing a  column  of  air  under  varying  conditions.  If  a  column  of  air  between 
two  levels  is  of  uniform  temperature  and  perfectly  dry,  we  can  easily 

log.  A 

determine  the  height  from  the  formula  h  =  -  ;  —  ^'    .  (i).  where  w  is  the 

i        /     .   w\v  " 
log.  (,  +  -) 

weight  of  unit  volume  of  air  under  the  conditions  of  temperature  and  pres- 
sure at  the  bottom  (or  top).  The  air,  however,  is  never  dry  and  for  con- 
siderable differences  of  level  is  never  of  uniform  temperature.  Consequently 
we  shall  have  to  break  up  our  column  into  columns  small  enough  to  be  con- 
sidered reasonably  uniform  throughout.  Then  by  adding  the  heights  of  our 
sections,  we  get  the  total  height  between  the  two  extreme  levels. 

Regnault  determined  the  weight  of  a  liter  of  dry  air  at  Paris  under 
standard  conditions  as  1.293233  grammes,  and  the  weight  of  a  liter  of 
mercury  as  13,596  grammes.  Hence  the  weight  of  a  cubic  foot  of  dry  air  at 

Paris  under  standard  conditions  is        93233  X  12,  expressed  in  inches  of 

i3596 
mercury. 

If  the  air  still  remains  dry  but  has  a  temperature  and  pressure  other 

than  the  standard,  the  weight  will  be    r'293233  x  12  X  —  X 

I3596  A       i 


where  t  is  the  temperature  centigrade.  If  the  air  contains  aqueous 
vapor,  we  can  consider  it  as  consisting  of  two  parts,  viz.,  a  cubic  foot  of 
dry  air  at  temperature  t  and  pressure/  —f,  and  a  cubic  foot  of  aqueous 
vapor  at  temperature  /  and  pressure  f,  where  f  is  the  tension  of  the 

aqueous  vapor.      The  weight  of  the  former   is   1'293233  x  I2  x  P  ~  f 

!3596  A 


32  THE  ATMOSPHERE 

X  — ; 1—7 — :,    and  the   weight   of   the   latter  f  x   1-293233  x  I2  x  /. 

i  +  . 00367  *'  13596  p0. 

X  — ; -, — -.     Hence,  the  weight  of  a  cubic  foot  of  moist  air  at  Paris 

i  +  .00367  t 

j    •       •        i  e  •      1.2T.QT.2T.T.  p  —  4  f  •  I 

expressed  m  inches  of  mercury  is—  X  12  x  -  -  X 


13596  29.9212       i +.00367* 

p    SL     f 

=  — T —     i     +  X  -0000381477. 
i  -j-  -00367  t 

It  will  be  seen  that  a  change  in  the  value  of  gravity  will  leave  the 
numerator  and  the  denominator  of  formula  (i)  practically  unaffected. 

The  barometric  coefficient  is 


( 

'  V 


i    I 

r 


.Q00038I477V 


.00367^         p 


The  following  example  will  serve  to  illustrate  the  method.  Two  points 
on  the  side  of  a  hill  were  by  leveling  determined  to  have  a  difference  of 
height  of  exactly  fifty  feet.  The  barometer  at  the  lower  point  read  30.2784. 
and  at  the  upper  point  30.224. 

Lower  point  Upper  point 

30.2784  30.224 

—.085    reduction  to  o°  C.  —.085  reduction  to  o°  C.. 

3°-I934  3o-i39 

-{-.037    correction  for  capillarity          +-037 

30. 2304  corrected  reading  lower  point  30. 176  corrected  reading  upper  point 
f-^/was  found  to  be  .091  inch,  and  i  -j-  .00367  t  was  1.055. 

A  — 1/=  30-1?6  —  .091  =  30.085. 

Log.  30.085  =  1.4783500. 
Log.  1.055  =  0.0232525. 
Log.  .0000381477  ='5.5814485. 

P  —  %  f  — 

''•  L°g'  i  +   00367  t  X  •000°38l477  =  3-0365460. 

This  logarithm  corresponds  to  .0010877  inch.  This  shows  that  under 
the  conditions  a  cubic  foot  of  air  weighed  .00108  inch  of  mercury. 

p  -\-  .001087  =  30.1771 
log.  30.1771  =  1.4796775 
log.  30.176    =  1.4796617 

.0000158 


HEIGHT  35. 

Hence,  K  = =  63291.  i 

.0000158 

lo£-  A  =  1-4804438 
log-  A  =  i-47966l7 

.0007821  x  K—  49.499  feet. 

Such  short  distances  are  a  severe  test  for  a  barometer.  Where  the 
heights  are  greater  the  proportional  error  should  be  less.  By  no  refinement 
is  it  possible  to  measure  with  a  barometer  closer  than  a  foot,  since,  as  we 
have  seen,  the  weight  of  a  cubic  foot  of  air  is  equivalent  to  .001  (-(-  or  — ) 
inch  of  mercury  and  barometers  do  not  read  finer  than  thousandths  of  an 
inch.  By  careful  work  it  is  probable  that  the  error  need  not  exceed  five  feet 
in  a  thousand.  By  the  method  of  multiple  stations,  the  errors,  which  are 
as  likely  to  be  above  as  below,  will  be  apt  to  eliminate  themselves  in  the 
summation.  The  altitude  of  Mont  Blanc  was  computed  by  Delcros,  from 
barometric  measurements  taken  by  Bravais  and  Martins,  to  be  4810  meters, 
which  came  strikingly  close  to  the  result  obtained  by  a  geodetic  survey,  viz., 
4809.6.  But  it  was  a  mere  coincidence,  as  Delcros'  formula  was  not  suscep- 
tible of  such  accuracy,  since  it  wholly  neglects  the  moisture  of  the  air. 


MOTION  RELATIVE  TO  THE  EARTH 


VERTICAL    MOTION 

BEFORE  taking  up  the  subject  of  the  general  circulation  of  the  atmos- 
phere, it  will  be  necessary  to  consider  the  effect  of  the  rotation  of  the  earth 
upon  motion  at  or  near  its  surface.  We  shall  first  consider  the  effect  of 
this  rotation  upon  vertical  motion.  The  effect  of  the  rotation  of  the  earth 
upon  vertical  motion  has  little  practical  bearing  upon  meteorology,  whereas 
the  effect  of  this  rotation  upon  horizontal  motions  is  of  the  utmost  impor- 
tance. For  the  sake  of  completeness,  however,  we  shall  consider  the  former. 

Rotations  may  be  compounded- and  resolved  in  the  same  manner  as 
simple  forces.  Thus  the  rotation  of  the  earth  about  its  axis  can  be  resolved 
into  partial  rotations  about  any  set  of  axes,  and  the  partial  rotations  about 
these  axes,  taken  usually  mutually  perpendicular,  together  make  up  the 
equivalent  of  the  original  rotation. 

Thus,  at  any  point  of  the  earth's  surface,  we  can  draw  a  vertical,  which 
we  will  call  the  axis  of  z,  a  tangent  to  a  meridian  which  we  will  call  the 
axis  of  x  and  an  axis  mutually  perpendicular  to  these,  which  we  will  call  the 
axis  of  y.  If  we  consider  the  point  at  rest,  the  effect  of  the  earth's  rotation 
is  precisely  the  same  as  if  the  rotation  about  the  axis  of  z  were  GO  sin  A, 
where  A.  is  the  latitude  and  GO  the  angular  velocity  of  the  earth  about  its  axis. 

This  rotation  is  in  the  northern  hemisphere  counter  clockwise ;  in  the 
southern  hemisphere  clockwise.  The  rotation  about  the  axis  of  x  is  GO  cos 
A,  and  this  rotation  is  clockwise  in  the  northern  hemisphere.  There  is,  of 
course,  no  rotation  about  the  axis  of  y  for  the  same  point.  We  can  easily 
see  why  this  is  so  by  placing  a  system  of  axes  such  as  we  have  described 
on  a  globe,  and  then  by  moving  it  along  a  parallel  of  latitude  to  the  east- 
ward, noting  how,  to  keep  it  in  the  proper  position,  it  is  necessary  to  keep 
turning  the  vertical  axis  in  a  counterclockwise  direction,  while  the  axis  of  x 
must  be  turned  clockwise.  At  the  equator  we  turn  only  the  axis  of  x,  since 
here  sin  A  =  o  and  cos  A  =  i,  while  at  the  pole  we  turn  only  the  axis  of  z, 
since  here  sin  A  =  i  and  cos  A  =  o. 

Using  the  present  system  of  coordinates,  let  us  suppose  that  we  drop  a 
particle  from  a  height  h,  and  reckoning  the  point  directly  under  it  as  the 
origin  consider  y  positive  when  measured  to  the  westward.  If  the  earth 
were  at  rest  it  would  strike  at  the  point  directly  beneath;  but  actually  the 


MOTION  RELATIVE  TO  THE  EARTH  35 

earth  is  revolving  about  the  axis  of  x  with  a  clockwise  rotation  equal  to 

GO  COS  A. 

Let  us  first  suppose  that  the  direction  of  gravity  does  not  change  during 
the  fall.  At  starting  the  body  had  a  velocity  relatively  to  o,  the  origin  of 
coordinates,  of  h  <a  cos  A. 

If,  during  the  time  of  the  fall,  the  body 
would  have  described  a  small  arc  z  a,  then  on 
striking  it  would  be  at  a  distance  o  b  from  o 
equal  to  the  arc  s  a.  But  the  direction  of 
gravity  is  continually  changing.  The  rotation 
about  o  z  does  not  change  the  direction,  but 

the  rotation  about  o  x  does,  and  in  the  time  t        

turns  this  direction  through  an  angle  GO  cos  A./.  °       6 

There  is,  therefore,  superinduced  a  westward  FIG.  7 

component  to  gravity,  proportional  to  the  time, 
and  equal  to  g  sin  (GO  cos  A.  £),  which,  since  the  angle  is  small,  we  can  write 

g  GO  cos  A.  t. 

f  a 
The  horizontal  velocity  is,  therefore^oocos  A. h  GO  cos  A.     When 

the  particle  reaches  the  ground,  h  =  — — ,  so  that  the  horizontal  velocity  is 

2 

always  eastward  except  just  at  the  instant  of  striking,  when  it  is  zero.  The 
body,  therefore,  strikes  vertically.  There  is  practically  no  northerly  or 
southerly  component,  so  that  the  body  strikes  to  the  eastward  at  a  distance 

g  GO  cos  A. hco  cos  A.  t  or  %  g  GO  cos  A.  /",  from  the  point  vertically 

under  it. 

[As  a  matter  of  fact,  there  is  an  extremely  slight  motion  to  the  south- 
ward, owing  to  the  rotation  of  our  system  of  axes  about  the  axis  of  z  in  a 
counterclockwise  direction.  Careful  experiments  show  that  objects  dropped 
from  a  height  show  a  constant  easting  which  agrees  well  with  theory.  The 
deviation  for  one  hundred  feet  is  only  about  one-eleventh  of  an  inch,  but  by 
repeated  droppings  such  a  small  quantity  can  be  brought  out.  The  southing 
is,  of  course,  too  small  to  be  detected.] 

If,  on  the  other  hand,  we  suppose  a  body  to  rise  under  a  flotative  force, 
such  as  a  balloon  or  a  mass  of  hot  air,  to  a  height  h,  it  will  have  a  horizontal 

velocity  h  GO  cos  A  —  f  GO  cos  A.  — ,  where  /  is  the  ascensional  force,  supposed 

to  remain  constant.  Since  h  =  \  /.  f2,  the  horizontal  velocity  at  all  times 
is  zero.  In  other  words,  the  westerly  lag  is  exactly  overcome  by  the  easterly 
component  of  ascensional  force  due  to  its  change  of  direction,  so  that  the 
balloon  or  hot  air  would  rise  vertically,  or  always  remain  over  its  starting 
point,  provided  the  air  were  perfectly  still.  J.  M.  Bacon  states  that  on  one 


36  THE  ATMOSPHERE 

occasion,  in  perfectly  still  air,  his  balloon  rose  perpendicularly  and  "remained 
hovering  over  the  Crystal  Palace  grounds,  and  apparently  over  the  same 
spot  in  the  grounds,  for  some  twenty  minutes,  till,  as  altitude  increased,  the 
whole  enclosure  had  to  all  appearances  shrunk  to  the  dimensions  of  a  toy 
model." 

A  falling  balloon  or  a  falling  mass  of  air  would  fall  directly  to  the 
eastward,  and  if  we  suppose  the  resistance  of  the  air  to  be  proportional  to  the 
square  of  the  velocity,  or  r  =  Cvz,  this  deviation  to  the  eastward  would  be 

i  GO  cos  A.  g  t3  [  i ). 

V  4     / 

If  a  projectile  were  shot  directly  upwards  to  a  height  h,  it  would  ex- 
perience besides  the  westward  lag,  an  acceleration  to  the  westward  due  to- 
the  change  of  direction  of  gravity.  The  relative  westward  velocity  would, 

/ * 
therefore,  be  h  GO  cos  A  -f-  g  GO  cos  A  — .     Since  at  the  summit  h  =  %  gt*> 

where  t  is  the  time  of  ascent  or  descent,  the  relative  westward  velocity  at  the 
summit  is^  GO  cos  A.  t~.  The  distance  to  the  westward  at  the  summit  due 

t* 

solely  to  the  lag  is  h  GO  cos,  A  t,  and  that  due  to  the  acceleration  is  g  GO  cos  A.  -. 

Therefore,  the  total  westing  at  the  summit  is  f  g  GO  cos  A.  f.  The  pro- 
jectile, therefore,  starts  falling  at  a  distance  f  g  GO  cos  A  /".  to  the  westward 
of  the  vertical,  and  with  a  westward  velocity  of  g  GO  cos  A  f. 

If  it  were  at  rest  at  the  summit,  it  would  gain  £  g  GO  cos  A  /3  to  the 
eastward  on  striking,  and,  therefore,  would  be  this  distance  to  the  west.  But 
the  westward  velocity  would  carry  it  g  GO  cos  A  t3  to  the  west  of  the  vertical. 

.   GO  cos  A  F3  /* 
Therefore,  the  total  westing  is  f  g  GO  cos  A  t"  or  -|  —   —5 ,  where 

V  is  the  initial  velocity  upwards.  This  value  is  given  by  Laplace  in  the 
"Mecanique  Celeste,"  IV.,  p.  341.*  By  the  formula  given  above  for  de- 
scent under  resistance  the  easting  is  \  GO,  cos  A.  g  f  (  i = — I,  where  /  is. 

V  4     / 

the  time  of  descent.     When  the  descent  is  slow  this  becomes  converted  into 

a  westing.  Hence,  ashes  ejected  from  a  volcano  into  still  air  would,  on 
reaching  their  highest  point,  be  to  the  westward  of  the  volcano  and  have  a 
westward  velocity.  Even  if  at  rest,  they  would  fall  to  the  westward,  and 
this  westing  is  further  increased  by  the  initial  westward  velocity  at  the 
beginning  of  the  fall. 

As  an  example,  let  us  suppose  a  bullet  fired  upwards  at  the  equator  with 
an  initial  velocity  of  2,000  ft.  per  second.  The  resistance  of  the  air  is  neg- 
lected. The  times  of  ascent  and  descent  are  62.5  seconds,  while  the  height 

*  Ferrel  gives  for  the  easting  when  a  body  is  dropped  from  a  height  f  g  a  cos 

A  /8,  and  for  the  westing  of  a  projectile  shot  upward  $-  -  — , ("On  the. 

Motion  of  Fluid  and  Solid  Bodies  Relative  to  the  Earth  ").  Both  are  incorrect. 


MOTION  RELATIVE  .TO  THE  EARTH 


37 


reached  is  62,500  ft.  At  its  highest  point  the  bullet  is  372.65  ft.  to  the 
west  of  the  vertical  and  745 . 3  ft.  to  the  west  on  striking.  The  velocity  to 
the  westward  at  the  summit  is  nine  feet  per  second,  or  more  than  six  miles 
per  hour. 

Again,  neglecting  the  resistance  of  the  air,  let  us  suppose  that  the  vol- 
cano Krakatoa,  during  the  great  eruption  of  1883,  drove  its  contents  up- 
wards with  an  initial  velocity  of  6272  ft.  per  second,  which  is  about  twice 
the  highest  velocity  attained  for  projectiles.  Some  of  the  particles  would 
reach  a  height  of  116  miles,  or  twice  the  height  of  the  atmosphere.  At 
the  highest  point  they  would  have  a  westward  velocity  of  88  ft.  per 
second,  or  60 'miles  per  hour.  The  time  of  ascent  would  be  196  seconds. 
At  the  highest  point  they  would  be  11,498  ft.  to  the  westward,  or  more 
than  two  miles.  If  we  consider  the  gases  merely  of  the  explosive  wave, 
which,  as  we  shall  see  later,  may  be  propagated  with  very  great  veloci- 
ties, these  would  be  quickly  condensed  into  solid  particles  (SO2  and  H2O), 
which  on  striking  the  atmosphere  on  their  return  would  have  their  velocities 
destroyed  by  the  resistance  of  this  medium,  although  passing  upward  through 
the  air  with  the  wave  they  would  not  experience  this  resistance. 

Stokes  has  given  the  following  formula  for  the  velocity  of  particles 

2   (r     iff  \ 

falling  through  a  viscous  medium.     V  —  — ^~,  I i  I  a*,  where  ff  and  p  are 

9  A4    \P         / 

the  densities  of  the  particles,  and  the  medium  respectively,  a  is  the  average 
radius  of  the  particles  and  ^,  is  what  Stokes  calls  the  ' '  Index  of  Friction. " 

This  index  of  friction  is  defined  by  the  equation  ^l=  — ,  where  //  is  the  co- 

ft 

efficient  of  viscosity.  //  is  taken  as  .116"  for  air.  From  this  formula  the 
following  table  is  derived. 


Radius  of  Particles  in  Inches 

Feet  per  Day 

Time  of  Falling  50,000  Feet 

.00003 
.00007 

68 
368 

2  years 
136  days 

As  the  atmosphere  did  not  become  clear  for  two  years  after  the  eruption 
of  Krakatoa,  it  is  inferred  that  the  particles  averaged  about  .00006  inch 
in  diameter,  which  agrees  well  with  other  theoretical  deductions  based  upon 
the  optical  phenomona  of  Coronae,  etc. 

The  above  treatment  of  relative  vertical  motion  is  that  of  Routh  ("Ad- 
vanced Dynamics").  As  it  does  not  take  into  consideration  the  fact  that  the 
moment  of  the  velocity  about  the  earth's  axis  remains  constant,  it  is  merely 
an  approximation,  though  when  the  distances  are  small  a  very  close  one. 


38  THE  ATMOSPHERE 

According  to  the  formulas  above  it  will  be  seen  that  the  body  when  dropped 
from  aloft  strikes  the  earth  vertically  to  the  eastward,  while  when  projected 
upward  and  falling  back  again  it  has  a  westward  velocity  on  striking. 

Strictly,  on  account  of  the  conservation  of  areas,  the  body  on  falling 
back  after  being  projected  upward  must  strike  vertically,  while,  when 
dropped  from  aloft,  it  must  have  an  eastward  velocity  or  striking.  Still, 
for  short  distances,  these  velocities  are  so  slight  that  the  formulas  given 
above  are  nearly  correct. 

A  rigorous  treatment  is  as  follows.  Since  the  body  when  projected 
upward  or  dropped  from  aloft  always  remains  in  a  fixed  plane  which  passes 
through  the  center  of  the  earth  and  is  tangent  to  the  parallel  of  latitude  at 
the  initial  point,  the  body  in  either  case  falls  south  of  the  starting  point  in 
the  northern  hemisphere.  If  we  suppose  the  body  dropped  from  a  height  h, 
so  that  R  -f-  h  =  R',  where  R  is  the  radius  of  the  earth,  and  q>  is  the  angle 
in  the  fixed  plane  between  the  radius  vector,  r,  at  any  time  and  its  initial 

position,  then  cp  =  —     —  5  --  .     This  follows  from  the  law  of  the  con- 

servation of  areas.  We  shall  throughout  adopt  the  Newtonian  notation  of  a 
super  dot  for  velocities  and  a  double  super  dot  for  accelerations. 

R'  "  GD  cos  /I       ^  „.       2  R1 
Hence,  <p  =  -—  .,.„•     Calling  —  -,  a\  we 

• 


v 

?'  *  63  COS  A, 

1      -    J  1 

^•a 

„.  _ 

e  („•  _  /•)•  -  ^  L(«  +  ty  +  (a  +  t)  (a-  t)  +  (a-ty\  * 

R1  cos  A  co    r  I"       i  2  i       "I    , 

can  write  g>  =  —       -    /   I  -,  —  r—r^  +  /  —  ,   ...  ,  -  -^  4-  -,  -  T-TTJ  I  a  t. 
2  g        J    L(a  +  ty  T   (a+t)  (a  —  t)       (a-  /)'  J 

n          _  R1  oo  cos  1  r      i  i  i  .         (a  —  t) 

-~~      :~~     g' 


Neglecting  the  small  southing,  the  easting  is  R  (cp—  cos  A.,  oot).  As  a  prac- 
tical example  let  us  suppose  that  a  body  falls  from  the  height  of  a  mile  at  the 
equator.  The  time  is  18  seconds  and  a  is  found  to  be  1143.77.  Substituting 
these  values,  we  get  as  thevalueof  q>,  <p=.  0013092048.  Since  the  radius  of 
the  earth  at  the  equator  is  20,926,202  ft.,  R'  =  20,931,482.  R  q>  —  27,396.68 
ft.  The  earth  has  advanced  1521.753  ft.  per  second  or  27,391.56  ft.  in  the 
time  of  fall.  Hence,  the  easting  is  R  (<p  —  cot)  =  5.  12  ft.  Where  the 
height  is  considerable  and  the  southing  appreciable,  we  may  determine  both 
the  easting  and  southing  by  solving  a  spherical  triangle.  The  easting  given 
by  the  approximation  formula  of  Laplace  and  Routh,  viz.,  ^  g  GO  cos  \  t\  is 


MOTION  RELATIVE  TO  THE  EARTH 


39 


4.52  ft.    For  lesser  heights  the  values  given  by  the  two  formulas  are  prac- 
tically identical. 

HORIZONTAL    MOTION 

We  have  now  to  consider  the  important  problem  of  the  effect  of  the 
rotation  of  the  earth  upon  bodies  moving  horizontally  or  parallel  to  its  sur- 
face. The  motion  is  supposed  to  take  place  without  any  friction.  We  shall 
first  consider  the  hypothetical  case  that  the  earth  is  a  sphere  and  then  the 
actual  case  of  a  spheroid.  To  fix  our  ideas,  we  shall  suppose  a  smooth  ball 
rolling  over  a  perfectly  smooth  sphere  of  the  size  of  the  earth.  We  must 
distinguish  between  the  absolute  motion  of  the  body  and  its  motion  relatively 
to  the  earth  when  the  latter  is  rotating  on  its  axis.  It  will  be  seen  that  it 
is  impossible  for  the  body  to  maintain  a  position  relatively  at  rest  to  the 
earth,  except  at  the  equator.  Since  there  is  absolutely  no  interaction  be- 
tween the  body  and  the  earth  except  in  a  direction  at  right  angles  to  the 
motion,  the  body  being  given  an  initial  absolute  velocity  in  a  certain  fixed 
plane  will  always  remain  in  that  plane,  describing  in  space  a  fixed  great 
circle.  The  absolute  velocity  will  remain  constant  and  the  path  described 
on  the  rotating  earth  will  be  a  composition  of  the  motion  of  the  earth  and  the 
motion  of  the  body  in  the  fixed  great  circle. 

The  angular  velocity  of  the  earth  is  .2618  radian  per  hour.  If  we 
suppose  the  body  projected  with  a  velocity  of  eighty  miles  an  hour  due 
east  at  30°  Lat,  it  will  describe  a  fixed  great  circle  tangent  to  the  parallels 
30°  N.  and  30°  S.,  crossing  the  equator  at  points  equidistant  from  these  two 
extremes.  The  velocity  of  the  earth  at  this  latitude  is  898  statute  miles, 
since  the  velocity  at  the  equator  is  1037  miles.  The  absolute  velocity  of 
the  body  is  978  miles. 

At  the  end  of  an  hour  the  point  from 
which  it  started  on  the  earth's  surface  will 
be  898  miles  to  the  eastward  on  the  same 
parallel  of  latitude.  The  body,  however, 
will  be  on  the  great  circle  A  C,  at  a  dis- 
tance A  C  equal  to  978  miles.  The  angle 
A  P  B  =  15°.  The  angle  A  C  =  14°  9', 
since  we  take  the  radius  of  the  earth  to  be 
3958  miles,  and  the  circumference  24,869 
miles.  By  spherical  trigonometry  we  find 
that  the  angle  A  P  C  =  16°  14'.  A  great 
circle  tangent  to  the  parallel  of  30°  at  B  is 
the  direction  of  due  east  at  B.  This  great 
circle  cuts  the  circle  P  C  at  a  point  in  FlG-  8 

latitude  29°  39',  while  C  is  in  latitude  29°. 

Consequently  the  body  has  deviated  to  the  right  39'  or  39  geographical  miles 
from  due  east,  and  is  moving  in  an  absolute  direction  8°  2'  south  of  east. 


Of   THE 

f   UNIVERSITY   ) 

Of 
irr\o*>l\K, 


4O  THE  ATMOSPHERE 

It  has,  therefore,  a  component  of  velocity  of  968.4  miles  due  east  and 
another  of  136^.7  miles  due  south.  The  earth  at  this  point,  C,  has  a 
velocity  of  907  miles.  Therefore  the  relative  eastward  velocity  will  be 
61.4  miles  instead  of  80  miles,  as  it  was  originally.  This  compounded 
with  the  southward  velocity  of  136.7  miles  gives  a  velocity  relatively  to 
the  earth  of  150  miles,  and  the  direction  relatively  to  the  moving  earth  is 
66°  south  of  east.  It  crosses  the  equator  at  an  absolute  angle  of  30°,  with 
an  absolute  eastward  velocity  of  847  miles  and  a  southward  velocity  of 
489  miles. 

Relatively  to  the  earth,  it  will  have  at  the  equator  a  westward  velocity 
of  190  miles.  Hence  relatively  to  the  earth  it  will  cross  the  equator  with  a 
velocity  of  524  miles  and  in  a  direction  68°  46'  south  of  west.  We  can  easily 
compute  that  after  two  hours  from  starting  the  body  will  be  in  Lat.  26°  7'  N. 
and  moving  with  a  velocity  relatively  to  the  earth  of  12  miles  to  the  east. 

Shortly  after  this,  just  below  Lat.  26°,  it  will  have  no  relative  east  or 
west  motion. 

After  three  hours,  it  will  be  in  Lat.  21°  39'  N.,  and  moving  with  a 
relative  west  velocity  of  53  miles,  and  with  a  southward  velocity  of  355 
miles.  The  total  relative  velocity  will,  therefore,  be  478  miles,  and  it  will 
he  moving  in  a  direction  81°  31'  south  of  west. 


30° 


FIG.  9 

It  will  be  seen  that  the  body  will  describe  on  the  moving  earth  a  path  IIKC 
that  in  Fig.  9,  moving  gradually  to  the  westward  and  oscillating  between 
Lats.  30°  N.  and  S.,  while  executing  a  loop  at  these  limits. 

If  the  body  were  started  without  any  relative  velocity,  that  is,  if  we 
imparted  to  it  an  absolute  velocity  of  898  miles  due  east  at  30°,  it  would 
•oscillate  between  the  parallels  of  30°  as  before,  but  the  loops  would  be 
changed  into  cusps.  If  we  imparted  to  it  a  relative  velocity  due  west,  it 
would  trace  on  the  moving  earth  a  wavy  curve  like  in  Fig.  10. 

As  another  example  let  us  suppose  that  the  body  is  started  due  west  at 
3°°  with  a  relative  velocity  of  98  miles.  Since  the  earth  here  has  a  velocity 


MOTION  RELATIVE  TO  THE  EARTH  41 

to  the  eastward  of  898  miles,  the  absolute  velocity  of  the  body  is  800  to  the 
east.  At  the  end  of  an  hour  it  will  be  in  Lat.  29°  2&,  and  will  be  moving 
in  an  absolute  direction,  i.e.,  a  direction  referred  to  stationary  coordinates 
of  6°  34'  south  of  east.  The  point  from  which  it  started  will  have  moved 


FIG.  10 

on  the  parallel  of  30°  a  distance  of  898  miles.  The  great  circle  tangent  to 
this  point  will  cut  the  meridian  on  which  the  body  now  is  in  Lat.  29°  59'. 
The  direction  of  west  lies  on  this  great  circle.  Consequently,  the  body  which 
was  projected  due  west  is  now  39  geographical  miles  to  the  left  of  west. 

Consequently,  it  is  not  true  that  a  projectile  always  deviates  towards 
the  right  in  the  northern  hemisphere. 

FRICTIONLESS  MOTION  OVER  A  ROTATING    SPHEROID 

We  now  come  to  consider  the  case  of  a  body  moving  without  friction 
over  a  centrally  attracting  spheroid.  The  surface  of  still  water  on  the  earth 
assumes  a  spheroidal  form  because  it  is  subjected  to  two  forces,  viz.,  gravity 
acting  downward  and  centrifugal  force  acting  in  a  direction  at  right  angles  to 
the  axis  of  the  earth  and  away  from  it.  The  resultant  of  these  two  forces  must 
be  everywhere  normal  to  the  surface  of  still  water  and  by  Clairaut's  Theorem 
such  a  level  fluid  surface  is  a  spheroid  with  the  minor  axis  through  the  poles. 
At  the  equator  the  centrifugal  force  is  about  ^TF  of  the  true  force  of  gravity, 
so  that  bodies  there  are  lighter  by  this  amount.  The  weight  of  a  body  at 
any  other  point  is  its  true  gravitation  diminished  by  the  component  of  the 
centrifugal  force  in  the  direction  of  gravity. 

We  shall  suppose,  as  before,  a  small  sphere  rolling  upon  a  perfectly 
smooth  spheroid  which  represents  the  earth.  It  will  now  be  possible  for 
the  body  to  remain  at  rest  relatively  to  the  earth.  All  that  is  necessary  is 
that  it  shall  have  the  same  velocity  as  the  point  on  which  it  rests.  There  is 
now  an  equilibrium  between  the  gravitational  and  centrifugal  forces,  and  a 
very  delicate  one  at  that.  The  slightest  difference  in  the  velocities  will  set 
the  body  in  motion  and  in  time  it  will  drift  far  from  its  original  position. 
We  shall  designate  motion  along  a  meridian  as  polar,  and  motion  along  a 


42  THE  ATMOSPHERE 

parallel  of  latitude  as  horizontal.     The  latter  term  is  chosen  for  want  of  a. 
better  one,  as  it  would  be  inconvenient  to  call  it  east-west  motion. 

If  now  the  body  moves  due  west  relatively  to  the  earth,  an  acceleration 
will  urge  it  towards  the  pole,  while  if  the  body  move  due  east  an  acceleration 
will  urge  it  towards  the  equator.  Let  us  suppose  that  the  body  is  moving 
due  west  at  Lat.  30°  with  a  velocity  of  eighty  miles  an  hour.  Denoting  the 
earth's  radius  by  R,  its  angular  velocity  by  OP,  and  the  angular  velocity  of  the 
body  about  the  axis  of  the  earth,  or  the  horizontal  angular  velocity,  by  fy, 
we  have  for  the  acceleration, 

f=R  sin  5  cos  5  (of  -  ^), 

a  positive  value  denoting  acceleration  towards  the  pole,  a  negative  value 
towards  the  equator. 

Since  GO  =  .2618  and  ^,  under  these  conditions  is  .2385,  we  have  GO*  — 
.06853  and  ip*  =  .0568,  GO*  —  ^  =  .0117,  sin  5  cos  $  =  .433  and/  =  20, 
where  the  acceleration  is  expressed  in  miles  per  hour.  Expressed  in  foot- 
seconds  it  is  .0082,  which  is  only  nnnr  °f  tne  acceleration  of  gravity. 

What  it  amounts  to,  is  that  the  atmosphere,  owing  to  the  rotation  of  the 
earth,  is  whirled  away  from  the  pole  against  gravity,  bulging  out  until  it 
acquires  a  position  where  the  tangential  components  of  gravity  and  centrifu- 
gal force  balance  each  other.  This  equilibrium,  as  we  have  said,  is  a  very 
delicate  one,  and  when  the  proper  velocity  about  the  earth's  axis  is  not 
reached,  the  air  falls  again  towards  the  pole.  In  the  case  we  have  just  con- 
sidered of  a  wind  due  west  with  a  velocity  of  eighty  miles  an  hour  at  30° 
Lat.,  the  air  is  urged  back  towards  the  pole  at  the  rate  of  twenty  miles 
an  hour. 

Since  the  force  is  a  central  one,, 
the  areas  described  by  our  body  about 
the  axis  of  the  earth  in  equal  times 
must  always  be  equal,  or  the  law  of 
conservation  of  areas  must  hold. 

Let  us  suppose  we  are  looking 
down  on  the  spheroid  as  in  Fig.  n. 
The  pole  P  is  in  the  center.  If  now 
we  start  our  body  at  A  with  a  hori- 
zontal angular  velocity,  \j/,  less  than  &>, 
the  angular  velocity  of  the  earth,  it 
will  be  urged  poleward  and  will  de- 
scribe a  spiral  about  the  axis  repre- 
sented  by  the  path  A  B.  By  the  law 
of  conservation  of  areas  its  angular  velocity  will  increase  as  it  gets  nearer 
to  the  axis  and  will  finally  be  greater  than  GO,  which  remains  constant. 
The  body  will,  therefore,  oscillate  between  two  extreme  latitudes.  That 


MOTION  RELATIVE  TO  THE  EARTH  43 

is,  it  will  at  first  be  urged  towards  the  pole  and  then  its  velocity 
increasing  with  the  latitude,  it  will  be  actuated  by  an  acceleration  in  the 
reverse  direction,  which  will  first  overcome  its  inertia  towards  the  pole 
and  finally  drive  it  back  again  to  the  latitude  from  which  it  started.  At 
a  certain  point  in  its  path  it  will  have  precisely  the  horizontal  angular 
velocity  of  the  earth,  and  at  this  point  will  suffer  no  acceleration  in  either 
direction.  The  latitude  where  this  occurs  we  have  represented  by  the  circle 
C,  and  we  shall  call  this  circle  the  circle  of  equilibrium.  The  circle  of 
equilibrium  may,  therefore,  be  considered  the  circle  of  mean  position.  The 
work  done  on  the  body  by  the  excess  force  while  the  body  is  outside  the  circle 
creates  an  extra  velocity  which  is  destroyed  by  the  work  done  by  the  repel- 
lent force  in  pushing  it  out  of  the  circle  again. 

Calling  vh  the  horizontal  velocity,  since  the  moment  of  this  velocity 
about  the  earth's  axis  remains  constant,  we  have  cos*  #  ip  =  6,  where  #  is 
the  latitude.  Since  the  poleward  acceleration  is 

R  3  =  R  sin  3  cos  3  (GO* r 


Or,  3  =  sm  2  ^  o>'  -  C*  t&n[  3.  sec9  3. 


Multiplying  by  3  and  integrating, 

sm_2_3  3,.  $  _  £••  tan  $  seci 


cos  2  3  C  *        2  _,       „ 

—  =  --  a?  --  tan2  3  +  K. 

2  4  2 

2  50  -  cos  2  3  ^  +  c,  ^  _  tan.  ( 


Since  cos*  3-  =  —  ,  cos  2  30  —  cos  2  $  =  —  ---  — 
0  ^o        ^ 

and  tan8  30  -  tan*  3  =  ^  (j>0  -  ^). 

We  can  therefore  write  Equation  (i) 


GO* 

Two  values  of  ^»  make  3-  vanish,  viz.  ,  t/»  =  ipoy  and  ^  =  -r—  .      The 

& 

latter  value  is  the  extreme  upper  value  of  ^,  which  we  shall  designate  by 
^tt.    It  is  thus  seen  that  the  angular  velocity  of  the  earth  is  the  geometric 


44  THE  ATMOSPHERE 

mean  between  the  greatest  and  least  horizontal  angular  velocities  of  the 
body,  corresponding  to  the  highest  and  lowest  latitudes  which  it  reaches. 
From  (2)  we  have, 


where  vp  is  the  polar  velocity  and  vu  and  v0  represent  the  extreme  horizon- 
tal velocities,  while  vh  is  the  horizontal  velocity  at  any  intermediate  point. 
It  will  be  seen  that  the  maximum  polar  velocity  occurs  when  the  hori- 
zontal velocity  is  a  geometric  mean  between  the  two  extreme  horizontal 
velocities  or  when  vh  =  ^v^  v<.  Writing  vpm  for  the  maximum  polar  velocity, 

we  have 

vPm  =  vu-v0        (4). 

The  horizontal  angular  velocity  at  this  point  is  likewise  the  geometric 
mean  of  the  two  extreme  horizontal  angular  velocities,  or  GO. 

This  occurs  on  the  parallel  of  equilibrium.  Here,  relatively  to  the 
earth,  the  horizontal  velocity  is  zero,  so  that  the  path  traced  by  the  body  on 
the  moving  earth  is  first  westward,  then  curves  round  towards  the  right, 
cutting  the  parallel  of  equilibrium  at  right  angles.  It  continues  to  curve 
to  the  right  until  it  goes  due  east,  where  it  reaches  its  highest  latitude.  After 
this  it  cuts  the  parallel  of  equilibrium  at  right  angles  going  due  south,  and 
turns  due  west  again  at  its  lowest  latitude. 

OB?" 

Since  cos'S0  i/>0  =  cos"  5M  tpu  and  ipu=  — ,  wehave  cos50^0=  cos  $u  GO  (5), 

0. 
or  the  absolute  velocity  at  the  lower  latitude  is  equal  to  the  velocity  of 

the  earth  at  the  upper  latitude.  In  the  same  way  we  can  show  that  the 
absolute  velocity  of  the  body  at  the  upper  latitude  is  equal  to  the  velocity  of 
the  earth  at  the  lower  latitude. 

Thus  Equation  (5)  can  be  written: 

v0  =  veu  (6)  or  vu  =  veo  (7),  where  veo  and  veu  are  the  velocities  of  the 
earth  at  the  lower  and  upper  limits  respectively. 

From  these  equations  or  from  Equation  (i)  the  values  for  one  of  the 
limits  can  be  determined  when  those  of  the  other  are  known. 

It  will  readily  be  seen  that  the  path  of  the  body  as  traced  upon  the 
moving  earth  will  be  like  that  represented  in  Fig.  12.  The  body  will 
continually  oscillate  between  two  extreme  parallels  of  latitude,  cutting  the 
parallel  of  equilibrium  at  right  angles  and  moving  continually  towards  the 
west  while  executing  a  series  of  symmetrical  loops. 

As  an  example,  let  us  suppose  the  body  launched  due  west  at  10°  N. 
Lat.  with  a  velocity  relatively  to  the  earth  of  226  miles  an  hour.  The 
absolute  initial  velocity  is  1021  —  226  =  795. 


MOTION  RELATIVE  TO  THE  EARTH 


45 


The  initial  horizontal  angular  velocity  is  therefore  .  204  and  GO  =  .  2618. 


Since  ^  =  .204= 


COS 


we 


tnat  ^=- 


C*  =  .03914  and  a? 


=  .  06853.  We  find  that  the  superior  latitude  of  the  spiral  will  be  39°  57', 
as  the  earth  moves  here  with  a  velocity  of  795  miles.  (By  Equations  (6) 
and  (7).) 

The  circle  of  equilibrium,  where  there  is  no  polar  acceleration  and  the 
maximum  polar  velocity,  is  28°  53'.     Here  the  polar  velocity  is  226  miles 


226 


10 


FIG.  12 


10- 


and  the  horizontal  velocity  908,  the  same  as  that  of  the  earth.  At  the 
upper  limit,  39°  57',  the  horizontal  angular  velocity  is  .  3359  and  the  hori- 
zontal velocity  1021  miles.  Hence  relatively  to  the  earth  it  is  moving  here 
with  an  eastward  velocity  of  226  miles.  The  absolute  velocity  at  the  parallel 
of  equilibrium  is  929  miles.  Re- 
latively to  the  earth  the  velocity  is 
226  miles  at  all  points. 

If  the  body  had  been  launched 
at  10°  with  a  velocity  of  eighty 
miles  to  the  west,  the  superior  0- 
limit  would  be  about  25°  N.  Lat. 
and  the  parallel  of  equilibrium 
would  be  18°  55'.  If  we  should 
launch  it  to  the  east  at  10°  instead  ^ 
of  to  the  west,  there  are  three 
cases.  With  velocities  above  six- 
teen miles,  it  would  describe  a  path  like  Fig.  13,  oscillating  between 
10°  Lat.  N.  and  S.,  and  moving  gradually  east  or  west  according  to  its 
velocity. 

With  velocities  less  than  sixteen  miles  an  hour  it  would  remain  wholly 
in  the  northern  hemisphere  and  describe  a  path  like  the  first  one  figured. 
But  if  launched  with  a  velocity  of  exactly  sixteen  miles  it  would  curve 


FIG.  13 


46  THE  ATMOSPHERE 

around  onto  the  equator,  where  it  would  remain,  proceeding  west  with  a 
velocity  of  sixteen  miles. 

The  above  discussion  of  the  equilibrium  of  a  body  on  a  rotating  spheroid 
where  there  are  no  frictional  forces  applies  directly  to  the  equilibrium  of 
fluids  on  such  a  rotating  spheroid.  Supposing  the  fluid  to  be  throughout  of 
a  uniform  temperature,  it  can  only  be  at  rest  relatively  to  the  spheroid  under 
the  conditions  that  the  surface  of  the  fluid  assumes  the  form  of  an  ellipsoid 
of  revolution  and  that  each  particle  has  its  proper  velocity  which  is  a  function 
of  its  distance  from  the  axis.  So  delicate  is  this  equilibrium  that,  as  we  have 
seen,  a  comparatively  slight  change  in  the  velocity  of  a  particle  will  cause  it 
to  depart  widely  from  its  original  position,  to  which  as  a  rule  it  never  re- 
turns. If,  then,  we  were  to  heat  such  a  fluid  in  equilibrium,  the  atmosphere 
for  instance,  unequally,  thereby  setting  up  convection  currents,  it  will  be  seen 
that  these  currents  would  differ  widely  from  what  they  would  be  on  a  non- 
rotating  spheroid.  The  unequal  heating  would  give  rise  to  convective 
velocity,  but  this  velocity  would  immediately  disturb  the  delicate  equilibrium 
and  start  the  particles  upon  paths  such  as  we  have  recently  considered. 

If  we  have  a  fluid  contained  in  a  vertical 

>  circuit,  with  a  free  surface,  such  as  A  B,  and 

heat  the  column  A  above  the  column  B,  since 
all  fluids  expand  under  the  influence  of  heat, 
the  column  A  will  become  higher  than  B,  and 
under  the  influence  of  gravity  the  top  of  the 
column  will  fall  (flow)  towards  B.  There 
will  thus  be  set  up  a  circulation  of  the  fluid 
FIG.  14  from  A  to  B  in  the  direction  of  the  arrows, 

which   will   continue   as   long   as   there   is   a 

difference  of  temperature  between  the  two  ends.  We  shall  thus  have  a  heat 
engine  which  performs  the  work  of  setting  the  fluid  in  motion  and  overcoming 
the  friction.  When  the  heat  energy  used  up  is  exactly  equal  to  the  work  of 
friction,  the  motion  will  be  uniform.  Since  a  hot  column  of  air  weighs  less 
than  a  cold  column  of  equal  height,  the  pressure  at  the  bottom  of  A  must  be 
less  than  at  the  bottom  of  B.  It  is  thus  that  differences  of  temperature 
cause  differences  of  pressure  in  a  fluid  and  hence  motion. 

If  we  supposed  the  earth  at  rest  and  the  surface  everywhere  at  its  actual 
temperature,  then  there  would  be  a  flowing  away  of  the  air  from  the  equator 
at  its  upper  surface  towards  the  poles,  and  a  flowing  back  along  the  lower 
surface  towards  the  equator  again.  Since  the  mass  in  circulation  is  con- 
stant, and  the  available  space  about  the  poles  is  constricted,  the  velocity 
would  have  to  increase  with  the  latitude,  so  that  the  same  volume  might  cross 
different  sections  in  the  same  time. 

We  have  already  seen  that  from  the  law  of  conservation  of  areas 
=  C  see8  5. 


MOTION  RELATIVE  TO  THE  EARTH 


47 


Hence,       ip  =  2  C  sec.  *  3  tan  3  3  =  2  ^>  tan  3  3  and 

R  cos  $  ip  =  2  Rip  sin  3  3. 

Putting  ^r  for  the  horizontal  angular  velocity  relatively  to  the  earth, 
or  ^v  =  ip  —  GO,  we  have 

R  cos  3^=2^?  (i/>r  -f-  <*>)  sin  33  and  R  3 
=  —  R  sin  3  cos  $  (2  GO  ipr-\-  ^r2). 

Since  ^r  is  usually  small  compared  with  GO,  we  may  without  serious 
error  write  these  two  equations, 

R  cos  3  ip  =  2  R  GO  sin  3  3  and  R  3  =  —  2  ^  sin  3  cos  3  &?  ^r. 

The  square  root  of  the  sum  of  the  squares  of  these  quantities  is  the 
deflective  force  relatively  to  the  earth.  Consequently,  if  p  be  the  radius  of 
curvature  of  the  path  of  the  body  and  vr  the  relative  velocity  (total)  at 
any  point, 

R  A  /  4  &?2  sina  3  32  -f  4  sin2  3  cos2  3  a?2  i/>r*  =  ^-. 


=  2  co  sn 


Hence,  approximately,  p  = 


32  +  R*  cos2 
v. 


=  2  GO  sn 


v. 


and  the  angular  velocity  with 


2  GO  sin 
which  the  path  turns  is  2  GO  sin  5. 

This  result  was  first  given  by  Ferrel,  but  its  application  seems  to  have 
been  generally  misunderstood  by  meteorologists,  as  the  following  example 
of  what  is  found  in  some  text-books,  shows. 

"*If  a  body  be  supposed  to  move  without  friction  on  the  level  surface 
of  a  rotating  globe,  a  single  impulse  would  give  it  perpetual  motion ;  but  th"e 
motion  could  not  be  along  a  straight  path.  It  would  continually  be  deflected 
to  one  side  of  its  momentary  path,  to  the  right  in  one  hemisphere,  to  the  left 
in  the  other,  and  with  a  force  dependent  on  its  velocity  and  on  its  latitude ; 
but  independent  of  its  direction  of  motion.  Its  path  would  always  be  curved 
in  a  systematic  manner :  the  curvature  would  be  sharper  for  slow  motions 
than  for  rapid  motions,  and  sharper  in  high  latitudes  than  near  the  equator. 
The  following  table  will  give  some  idea  of  the  rate  at  which  a  moving  body 
tends  to  turn  from  a  straight  line  on  a  sphere  rotating  once  in  twenty-four 
hours. 

RADIUS    OF   CURVATURE    (IN  MILES)    FOR   FRICTIONLESS 
MOTION  ON  THE  EARTH'S  SURFACE 


Latitude    .     .     .     . 

0° 

5° 

10° 

20° 

30° 

40° 

5o° 

60° 

7o° 

80° 

90° 

20  miles  an  hour 

oo 

880 

442 

224 

153 

119 

IOO 

88 

82 

78 

77 

10  miles  an  hour 

oo 

440 

221 

112 

76 

59 

5° 

44 

4i 

39 

38 

5  miles  an  hour 

oo 

220 

110 

56 

38 

3° 

25 

22 

20 

19  |   19 

*W.  M.  Davis,  "Elementary  Meteorology." 


48  THE  ATMOSPHERE 

"A  body  once  set  in  motion  under  these  conditions  would  continue  mov- 
ing forever,  always  changing  its  direction  but  never  changing  its  velocity.  If  it 
were  given  a  velocity  of  twenty  miles  an  hour  in  any  direction  at  Lat.  30°, 
it  would  describe  a  series  of  overlapping  loops,  gradually  carrying  it  west- 
ward around  the  earth,  but  never  passing  outside  of  the  parallels  of  20° 
and  40°.  If  it  were  given  a  velocity  of  five  or  more  miles  an  hour  eastward 
at  Lat.  5°,  it  would  describe  a  scalloped  path,  oscillating  back  and  forth 
across  the  equator,  but  never  escaping  beyond  Lat.  5°  in  either  hemisphere." 

Some  of  this  is  correct,  but  most  of  it  is  not. 

A  body  moving  on  a  "globe"  or  "sphere"  suffers  no  deflecting  force  and 
its  path  is  not  that  described  in  the  above  quotation,  nor  is  its  relative  velocity 
constant.  A  deflecting  force  occurs  when  a  body  is  moving  over  a  spheroid 
and  is  due  to  the  contest  between  the  gravitational  and  centrifugal  forces. 
The  path  depends  greatly  upon  the  direction  in  which  the  body  is  launched. 
If  launched  with  a  velocity  of  twenty  miles  due  east  at  30°  Lat.,  it  would 
oscillate  between  30°  and  27°  43',  but  would  never  go  beyond  30°.  If 
launched  due  west  it  would  oscillate  between  30°  and  32°  9'  Lat.  If 
launched  due  north  or  south  it  would  oscillate  between  31°  and  29°. 

We  can  derive  the  result,  p  = -. — ^,  from  elementary  considera- 

2  GO  sm  3 

tions.  If  a  body  in  latitude  #  is  moving  over  the  earth  in  a  direction  A  B, 
withra  velocity  v  which  would  take  it  to  B  in  the 
time  d  f  if  the  earth  were  at  rest,  then  since  the 
earth  is  rotating  under  it  in  a  counterclockwise 
direction  in  the  northern  hemisphere,  instead  of 
arriving  at  B,  it  will  actually  arrive  at  C.  If  the 
center  of  curvature  of  this  path  is  at  O,  then,  since 
A  B  is  perpendicular  to  A  O,  it  follows  from  ele- 
mentary geometry  that  the  angle  C  A  B  is  half  the 
angle  CO  A.  The  arc  CA  =  vrdt  =  p  L  CO  A. 
/  C  A  B  =  GO  sin  #  dt.  =  %  L  C  O  A.  Hence, 

vr  FIG.  15 

~  2  sin  #  oa 

However,  this  is  merely  an  approximation,  as  we  have  pointed  out,  and 
does  not  hold  if  the  relative  velocity  becomes  excessive. 


GENERAL  CIRCULATION  OF  THE  ATMOSPHERE 


WITH  a  stationary  world,  hotter  at  the  equator  than  at  the  poles,  the 
circulation  would  be  along  the  meridians.  With  a  rotating  world,  the 
deflecting  force,  as  we  have  seen,  introduces  east  and  west  components, 
which  result  in  the  breaking  up  of  what  would  otherwise  be  a  simple  circula- 
tion into  practically  independent  fractions  separated  by  parallels  of  latitude. 
In  other  words,  instead  of  a  single  circulation  there  are  several  practically 
independent  circulations.  It  is  fortunate  that  this  is  so,  for  if  the  propelling 
forces  which  depend  upon  the  differences  of  temperature  were  united  along 
one  line  and  summed  together,  such  a  blast  would  result,  notwithstanding 
friction,  that  the  world  would  be  uninhabitable,  especially  in  the  higher 
latitudes. 

EQUATORIAL  CIRCULATION 

As  it  is,  we  have  to  consider  first  the  tropical  circulation  having  a 
motive  force  which  arises  from  differences  of  temperature  no  greater  than 
those  existing  between  the  equator  and  about  30°  Lat.  This  motive  force 
which  arises  in  the  vertical  components  of  the  circulation  is  further  reduced 
by  the  somewhat  adiabatic  expansions  and  compressions  which  it  under- 
goes. The  hot  air  arising  from  the  equator  loses  some  of  its  temperature 
by  expansion  and  the  return  descending  current  becomes  warmed  by  the 
descent,  so  that  finally  only  a  differential  result  is  effective  in  driving  the 
air  currents.  This  driving  force  is  about  .006  of  the  weight  of  the  air, 
while,  as  we  have  seen,  the  deflecting  force  is  about  -j-^-g-  or  .00025  of  the 
weight.  With  these  comparatively  slight  forces  the  circulation  of  the  atmos- 
phere is  carried  on  as  well  as  friction  overcome. 

The  motive  force  of  the  equatorial  circulations  is  thus  the  difference 
of  temperature  between  the  equator  and  some  point  to  the  north  and  south. 
Since  in  this  zone  the  surface  is  approximately  a  cylinder,  the  circulation 
can  take  place  in  an  approximately  free  manner.  That  is  to  say,  the  volumes 
of  the  circulating  air  at  the  two  extremes  will  not  be  markedly  changed,  as 
would  be  the  case  if  air  were  circulating  between  the  hotter  circumference 
and  the  colder  center  of  a  circle,  which  is  the  case  in  the  polar  circulation. 

Given,  therefore,  a  certain  motive  force  between  the  equator  and  a  point 
to  the  north,  depending  upon  their  difference  of  temperature,  it  is  evident 
that  the  resultant  velocity,  which  is  proportional  to  the  motive  force,  will 
develop  deflective  forces  which  will  determine  the  path  and  the  extent  of 
the  excursion  to  the  north.  If  the  extreme  point  to  the  north  coincides  with 
the  temperature  necessary  to  produce  the  driving  force  which  will  produce 


50  THE  ATMOSPHERE 

the  velocity  necessary  to  reach  it,  there  will  be  equilibrium.  If  at  the  point 
of  recurving  the  temperature  is  insufficient  to  produce  the  necessary  driving 
force  to  produce  the  requisite  velocity,  the  upper  border  of  the  path  will  fall 
back  until  it  finds  a  point  capable  of  providing  for  the  reduced  velocity.  If, 
on  the  other  hand,  the  temperature  potential  is  too  great,  the  velocity  will 
be  increased  and  the  upper  border  will  progress  upward  until  it  finds  a  point 
where  the  driving  force,  velocity  and  upper  border  are  in  adjustment. 

It  will  thus  be  seen  that  given  a  certain  distribution  of  temperature  on 
.a  planet,  the  width  of  the  equatorial  circulation  is  a  function  of  this  tempera- 
ture distribution  and  its  rotational  velocity. 

It  so  happens  upon  our  earth  that  a  distribution  of  temperature  (aver- 
age) ranging  from  27°  C.  at  the  equator  to  20°  C.  at  the  parallel  of  30°, 
is  just  sufficient  to  drive  (overcome  all  resistances  to)  the  air,  with  a  velocity 
sufficient  to  bring  it  up  to  the  parallel  of  extreme  temperature  which  is  about 
30°  Lat. 

The  shape  of  this  zone  permits  the  air  to  take  paths  approximately  like 
those  of  free  bodies,  though  this  is  not  exactly  the  case,  since  there  is  some 
constriction  at  the  upper  limit.  Such  a  circulation  will  necessarily  have  a 
•due  east  direction  at  the  upper  limit  and  a  due  west  direction  at  the  lower 
limit.  At  10°  Lat.  a  velocity  of  eighty  miles  an  hour  due  west  would  carry 
a  free  body  to  about  25°,  while  a  velocity  of  one  hundred  miles  an  hour 
would  carry  it  to  30°  before  it  recurved. 

We  know  that  velocities  like  these  are  constant  in  the  upper  layers  over 
the  equator.  This  fact,  previously  unsuspected,  was  revealed  by  the  great 
•eruption  of  Krakatoa,  when  a  gale  above  the  eight-mile  level  moving  with  a 
velocity  of  at  least  ninety  miles  an  hour  and  probably  more  at  the  highest 
levels,  carried  the  dust  in  a  due  west  direction  at  least  three  times  around  the 
globe.  Whymper  noted  during  an  eruption  of  Cotopaxi  (i°  15'  S.  Lat.) 
that  the  smoke  rose  vertically  in  still  air  until  it  attained  a  height  of 
40,000  ft.  (eight  miles),  when  it  encountered  a  powerful  current  blowing 
•due  west,  which  carried  it  rapidly  to  the  Pacific.  In  the  higher  levels,  from 
25°  to  30°,  the  currents  are  constantly  eastward,  generally  due  east,  with 
velocities  of  over  a  hundred  miles.  Thus  the  velocities  and  the  upper  limit  of 
the  equatorial  circulation  agree  well  with  what  we  should  expect  from  theory. 

Meteorologists  have  called  attention  to  the  fact  that  the  zone  comprised 
between  the  two  parallels  of  30°  contains  exactly  half  of  the  atmosphere, 
and  as  the  equatorial  circulations  are  roughly  measured  by  these  limits, 
some  connection  between  the  two  has  been  supposed  to  exist. 

But  as  we  have  already  pointed  out,  the  width  of  this  circulation  de- 
pends upon  the  temperature  distribution  and  the  deflecting  forces,  the  latter 
•depending  upon  the  rotational  velocity  of  the  earth. 

The  equatorial  circulation  of  the  planet  Jupiter,  if  we  may  judge  from 
the  equatorial  bands,  is  much  less  than  half,  which  is  probably  due  to  its 


GENERAL  CIRCULATION  OF  THE  ATMOSPHERE 


.great  rotational  velocity,  a   complete   revolution  being  performed   in  ten 
hours. 

We  shall  now  attempt  to  build  up  the  equatorial  circulation  from 
theory,  and  from  observation  as  far  as  this  is  possible  from  data  at  hand. 
We  shall  suppose  that  the  equatorial  stream  is  a  broad  band  extending 
10°  north  and  south  of  the  equator.  From  the  polar  borders  of  this  stream 
currents  start  out  at  a  high  level  with  the  velocity  of  the  stream  and  prac- 
tically due  west.  After  making  the  circuit  of  the  equatorial  circulation  these 
currents  return  into  the  main  stream  again  in  a  due  west  direction,  but  at  a 

C 


n  n  o 

s^   S>y  ^s^ 


FIG.  16 


lower  level.  Since  the  outgoing  and  incoming  masses  of  air  must  be  equal, 
the  level  separating  them  must  be  at  about  the  level  of  half  atmosphere, 
something  like  three  miles. 

We  shall  suppose  at  the  level  of  half  atmosphere,  therefore,  a  small 
flat  tube  A,  Fig.  16,  where  the  full  lines  represent  the  upper  outgoing  currents 
and  the  dotted  lines  the  lower  returning  current.  The  velocities  here  are  low 
and  the  polar  extension,  therefore,  small.  Over  this  small  tube  we  shall 
slide  the  larger  flat  tube  B.  The  outgoing  currents  here  start  from  a  higher 
level  and  the  incoming  currents  reach  the  main  stream  at  a  lower  level.  The 


FIG.  17 


FIG.  18 


velocities  are  also  higher  and  the  extension  north  correspondingly  greater. 
Over  this  tube  we  slide  the  larger  tube  C  and  so  on  until  the  highest  levels 
are  reached  for  the  outgoing  currents,  the  lowest  levels  for  the  incoming 
currents,  and  the  maximum  velocity  with  maximum  excursion  towards 
the  pole. 

The  vertical  section  of  these  tubes  would  be  something  like  Fig.  17. 
The  tubes  would  be  crowded  together  on  the  equatorial  border,  but  sepa- 
rated more  and  more  on  the  polar  border.  This  allows  the  volumes  to 
accommodate  themselves  to  the  moderate  constriction  towards  the  poles. 


52  THE  ATMOSPHERE 

At  the  equator,  on  the  surface  of  the  earth,  and  extending  to  each  side  for 
about  10°  is  a  wedge-shaped  space  of  calms,  the  borders  of  this  space  A 
being  limited  by  the  incoming  and  ascending  lower  currents.  The  height 
of  this  space  at  the  middle  is  about  eight  miles.  (Fig.  18.) 

The  equatorial  circulation  would,  therefore,  be  represented  by  a  series 
of  flat  tubes,  crowded  together  on  their  equatorial  border,  but  spreading  out 
on  their  polar  borders.  Each  tube  is  a  practically  independent  circulation 
where  the  currents  spiral  around  in  a  clockwise  direction  gradually  moving 
to  the  west.  The  core  would  carry  the  lowest  velocities,  while  the  velocities 
increase  from  the  core  outward.  This  is  necessitated  from  the  fact  that  a 
current  must  be  continuous,  not  being  capable  of  being  crossed  by  another 
current. 

Such  an  arrangement  satisfies  the  requirements  of  the  vertical  circula- 
tion, which  is  driven  by  the  differences  of  temperature  at  the  extremes.  It 
satisfies  the  conditions  of  continuity  and  the  stream  lines  are  such  as  would 
be  produced  by  the  deflecting  forces  due  to  the  earth's  rotation. 

We  must  next  inquire  how  such  a  theoretical  structure  agrees  with 
observation.  We  shall  call  the  incoming  lower  currents  the  trades;  the 
outgoing  upper  currents  the  antitrades. 

Just  beyond  10°  Lat.  we  should  find  the  trade  decreasing  with  the 
height  until  we  struck  the  beginnings  of  the  lowest  and  slowest  antitrades 
going  to  the  northwest.  This  has  been  observed  from  the  motion  of  clouds. 
At  a  little  higher  latitude  we  should,  after  passing  through  the  trades,  meet 
an  antitrade  blowing  first  to  the  east  and,  on  rising  higher,  one  blowing  to 
the  northeast  and  finally  one  blowing  to  the  north.  This  has  not  been 
observed  because  no  sounding  observations  to  great  altitudes  have  been  con- 
ducted in  low  latitudes.  At  present  we  do  not  know  whether  such  is  the 
fact  or  not,  but  for  the  benefit  of  our  theory  we  may  say  that  we  do  not 
know  that  such  is  not  the  fact. 

We  have  said  that  the  first  antitrades  met  with,  on  ascending,  should 
be  to  the  east.  This  is  confirmed  by  observations  in  low  latitudes.  The 
smoke  of  volcanoes,  after  breaking  through  the  trades,  has  repeatedly  been 
seen  to  go  to  the  eastward.  The  smoke  of  St.  Vincent  was  in  May,  1812, 
carried  eastward  to  Barbados.  The  smoke  of  Consequina  in  Nicaragua,  in 
T835,  went  E.  N.  E.  to  Jamaica.  Farther  north,  at  the  summits  of  Teneriffe 
and  Mauna  Loa  a  strong  S.  W.  antitrade  is  always  observed.  Humboldt 
states  that  at  the  top  of  Teneriffe  he  could  hardly  stand  before  this  south- 
west blast. 

In  Java  and  Sumbawa  smoke  from  volcanoes  has  drifted  east.  We 
have  thus  direct  evidence  of  currents  to  the  N.  W.,  N.  E.,  and  E.  as  required 
by  our  theory.  No  direct  observation  of  a  current  due  north  in  the 
antitrades  seems  to  be  at  hand,  although  of  necessity  such  a  current  must 
exist.  The  need  of  sounding  observations  for  the  lower  latitudes,  such  as 


GENERAL  CIRCULATION  OF  THE  ATMOSPHERE  53 

have  been  conducted  at  Blue  Hill  by  Rotch  and  Clayton,  and  in  Sweden 
is  thus  apparent.  The  upper  atmosphere  is  in  great  part  unexplored.  To 
sum  up  the  equatorial  circulation  seems  to  be  represented  by  a  curious 
system  of  flat  tubes,  the  outer  ones  containing  all  the  inner  ones,  yet  not 
concentrically.  The  stream  lines  in  each  tube  are  a  system  of  spirals  which 
progress  to  the  westward  and  have  for  their  horizontal  projections  the  series 
of  overlapping  loops  we  have  represented  in  the  figure.  The  vertical  pro- 
jections are  a  series  of  flat  ovals,  lying  completely  within  each  other  and 
crowded  together  at  the  equatorial  end.  The  conditions  fulfilled  by  such  a 
circulation  are  the  following: 

1.  The  extra  weight  of  the  polar  end  is  continually  forcing  a  current 
along  the  lower  level  to  the  equatorial  end,  where  it  rises  and  continually 
slides  back  again  to  the  polar  end  along  its  upper  level. 

2.  The  path  is  shaped  by  the  deflective  forces,  which  here  have  nearly 
free  play. 

3.  There  is  at  no  point  any  discontinuity.     That  is,  we  can  go  from 
one  point   to  another  by   insensible   gradations,   both   as   to   velocity   and 
direction  of  current. 

4.  Observations,  so  far  as  they  are  at  hand,  support  the  theory.      There 
are  at  present  no  contradictions. 

POLAR  CIRCULATION 

We  shall  next  consider  the  polar  circulation.  We  have  here  to  consider 
a  distribution  of  temperature  practically  on  a  plane,  a  circle,  which  changes 
gradually  from  the  circumference,  where  it  is  a  maximum,  to  the  center, 
where  it  is  a  minimum. 

The  general  circulation  here  must  be  of  the  nature  of  a  sinking  in  at 
the  center,  an  outflow  along  the  lower  levels  to  the  circumference,  where  it 
rises  and  gradually  makes  its  way  back  to  the  pole  over  the  upper  levels. 
The  current  directly  downward  near  the  center  must  be  considerable, 
although,  owing  to  the  spreading  out  of  the  current,  there  would  be  a  cone- 
shaped  space  of  calm  at  the  pole  itself.  As  the  stream  flows  off  from  the 
pole  it  would  acquire  a  westward  component  relatively  to  the  earth,  which 
would  develop  a  deflective  force  to  the  right,  tending  to  bring  it  back  to  the 
pole  with  a  short  turn.  But  the  deflective  force  here  does  not  have  free 
play.  It  is  opposed  and  overcome  at  all  points  by  the  outstreaming  air  from 
the  pole.  Consequently,  the  air  must  be  forced  out  ever  more  and  more, 
always  with  a  westward  component,  until  it  reaches  a  point  where  the 
deflective  force  is  at  last  equal  to  the  outward  pressure  gradient.  At  this 
point  it  circulates  directly  to  the  westward,  around  the  pole  in  momentary 
equilibrium.  But  it  now  rises,  becoming  warmer,  and  as  at  a  higher  level 
the  outward  gradient  is  weakened  (there  is  not  so  much  outflowing  air  here) 


54  THE  ATMOSPHERE 

the  stream  gradually  falls  back  towards  the  pole.  If  the  deflective  forces 
had  free  play  here,  the  paths  would  be  nearly  circular  and  of  short  diameter, 
but  the  inward  motion  is  checked  by  the  air  already  in  advance  seeking  its 
way  to  the  pole,  so  that  it  turns  towards  the  pole  very  gradually  instead  of 
sharply,  as  would  otherwise  be  the  case.  The  velocities  are  no  doubt  con- 
siderable at  the  upper  levels,  but  in  general  they  are  kept  in  check  by  friction 
on  the  outgoing  half  and  by  the  back  pressure  of  the  returning  half.  The 
important  point  is  that  the  deflective  forces,  although  very  pronounced,  are 
held  in  check  by  the  choking  forces  due  to  the  constriction  of  volume 
towards  the  center.  They  are,  therefore,  unable  to  shape  the  paths  of  the 
streams,  as  they  do  near  the  equator.  The  circulation,  therefore,  consists 
of  a  spiraling  outward  until  the  circle  of  equilibrium  is  reached  and  then  a 
spiraling  inward,  but  always  in  the  same  direction,  viz.,  to  the  westward. 
Of  course,  the  pressure  is  greatest  at -the  center  and  least  at  the  cir- 
cumference. 

The  work  done  by  a  heat  engine  is  proportional  to  the  difference  between 
the  two  extreme  temperatures  at  which  it  works.  Consequently,  the  friction 
overcome  by  the  polar  whirl  in  order  to  maintain  its  energy  of  rotation  con- 
stant is  proportional  to  the  difference  between  the  temperature  of  the  pole 
and  that  of  the  circle  of  equilibrium. 

This  ideal  polar  circulation  is  realized  only  approximately  in  the  case 
of  the  earth.  The  unequal  distribution  of  land  and  water,  especially  at  the 
north  pole,  is  a  greatly  disturbing  factor.  But  even  if  the  surface  of  the 
earth  were  homogeneous  at  the  poles,  owing  to  the  inclination  of  its  axis  and 
the  change  of  seasons,  the  pole  would  rarely  be  the  place  of  minimum 
temperature,  and  the  position  of  the  circle  (if  circle  it  were)  of  equilibrium 
would  be  extremely  variable.  Thus,  under  the  actual  conditions,  the  deter- 
mination of  the  polar  circulation  defies  all  analysis. 

Still  there  is  some  agreement  between  the  theoretical  case  we  have  con- 
sidered and  what  is  actually  found.  On  the  whole,  the  surface  winds  come 
out  of  the  pole  and  the  components  to  the  west  are  very  marked.  The 
particular  distribution  of  temperatures  at  the  earth's  poles  combined  with 
its  velocity  of  rotation  results  in  the  circle  of  equilibrium  lying  on  an  average 
about  25°  from  the  poles. 

From  the  homogeneity  of  its  surface,  the  south  pole  would  naturally 
make  a  nearer  approach  to  theory  than  the  north  pole.  We  quote  the  fol- 
lowing from  Ward  ("Climate")  : 

"The  rapid  southward  decrease  of  pressure,  which  is  so  marked  a. 
feature  of  the  higher  latitudes  of  the  southern  hemisphere  on  the  isobaric 
charts  of  the  world,  does  not  continue  all  the  way  to  the  south  pole.  The 
steep  poleward  pressure  gradients  of  these  southern  oceans  end  in  a  trough 
of  low  pressure,  girdling  the  earth  at  about  the  antarctic  circle.  From  here 
the  pressure  increases  again  towards  the  south  pole,  where  a  permanent 


GENERAL  CIRCULATION  OF  THE  ATMOSPHERE  55 

inner  polar  anticyclonic  area  is  found,  with  outflowing  winds  deflected  by 
the  earth's  rotation  into  easterly  and  southeasterly  directions.  A  chart  of 
the  south  polar  isobars  for  February  (after  Sir  John  Murray  and  Dr_ 
Buchan)  published  in  1898,  showed  a  pressure  of  29.00  inches  in  the  low 
pressure  girdle,  and  the  isobar  29.50  inches  around  the  inner  polar  area. 
These  easterly  winds  have  been  observed  by  recent  expeditions  which  have 
penetrated  far  enough  south  to  cross  the  low  pressure  trough.  The  limits 
between  the  prevailing  westerlies  and  the  outflowing  winds  from  the  poles, 
easterlies,  vary  with  the  longitude  and  migrate  with  the  seasons.  The 
change  in  passing  from  one  wind  system  to  the  other  is  easily  observed. 

"The  Belgica,  for  example,  in  latitudes  69^°-7i^°  S.  and  longitudes 
8i°-95°  W.,  was  carried  towards  the  west  by  the  easterly  winds  of  summer,, 
and  in  winter  was  driven  east  by  the  westerlies,  and  then  again  to  the  west. 
The  Belgica  thus  lay  in  winter  on  the  equatorial,  and  in  summer  on  the  polar 
side  of  the  trough  of  low  pressure.  The  seasonal  change  in  wind  direction 
was  very  marked,  being  almost  monsoon-like  in  character.  On  the  other 
hand,  the  English  expedition  at  77°  50'  S.  was  persistently  on  the  polar  side 
of  the  trough,  with  dominant  S.,  S.  E.,  and  E.  winds.  The  smoke  from 
Mt.  Erebus,  however,  showed  prevailing  southwesterly  currents.*  The 
German  expedition  on  the  Gauss  was  also  under  the  regime  of  easterly  winds 
during  its  .stay  in  winter  quarters.  The  Belgica  had  fewer  calms  than  some 
stations  nearer  the  pole.  The  south  polar  anticyclone  with  its  surrounding 
low  pressure  girdle  migrates  with  the  season,  the  center  apparently  shifting 
poleward  in  summer  and  towards  the  eastern  hemisphere  in  winter.  The 
cloudier  winds  are  poleward ;  the  clearer  winds  blow  out  from  the  pole.  The 
out-flowing  winds  from  the  polar  anticyclone  sweep  down  across  the  inland 
ice  and  are  usually  cold." 

Drygalski  found  foehn-like  winds  in  the  antarctic  regions,  blowing  a  gale 
mostly  from  S.  and  E.  It  will  be  seen  that  all  this  is  practically  a  descrip- 
tion of  such  a  circulation  as  we  have  deduced  from  theory. 

MIDDLE  CIRCULATION 

We  shall  lastly  consider  the  nature  of  the  circulation  between  the  polar 
circle  of  equilibrium  and  the  polar  border  of  the  equatorial  circulation.  We 
have  already  recognized  two  distinct  circulations,  viz.,  the  equatorial  circula- 
tion and  the  polar  circulation.  The  former  extends  over  something  like  30°,. 
while  the  latter  extends  over  something  like  25°.  We  shall  now  examine  the 
intervening  zone. 

At  the  polar  border  of  the  equatorial  circulation  the  currents  are  curv- 
ing more  sharply  under  the  influence  of  the  deflecting  force  than  at  any 
other  point.  They  are  here  going  directly  east.  The  neighboring  air  on  the 
outside  is  taken  along  with  them  by  friction  in  the  same  direction,  and,  there- 

*i.e.,  Moving  towards  the  S.W. 


THE  ATMOSPHERE 


fore,  tends  under  the  influence  of  the  deflecting  force  to  be  carried  towards 
the  equator.  But  the  equatorial  circulation  is  full,  incapable  of  receiving 
any  more  air.  These  outer  currents,  therefore,  exert  a  pressure  towards 
the  equator  without  being  able  to  move  in  that  direction. 

There  must,  therefore,  be  along 
this  border  a  ridge  of  high  pressure. 
In  Fig.  19  the  dotted  line  is  supposed 
to  represent  the  tipper  surface  of  the 
atmosphere  with  a  uniform  tempera- 
ture throughout  of  o°  C.  The  full 
line  represents  this  surface  under 
actual  conditions.  The  proportions 
are  greatly  exaggerated. 

The  surface  is  highest  at  the 
equator,  whence  it  slopes  down  grad- 
ually to  the  poles  where  the  height  is 
least.  At  the  polar  border  of  the 
equatorial  circulation,  A,  the  pressure 
is  greater  than  at  any  point  on  the 
earth's  surface.  At  the  pole  is  another 
place  of  maximum  pressure,  though 
FIG.  19  this  maximum  is  not  as  great  as  that  at 

A.  At  B.  the  border  of  the  polar  cir- 
culation is  a  place  of  minimum  pressure,  as  is  also  the  equator.  At  the 
border  of.  the  south  polar  circulation,  the  pressure  is  less  than  at  any  point 
on  the  earth's  surface.  The  minimum  at  the  border  of  the  north  polar  circula- 
tion is  about  the  same  as  that  at  the  equator,  perhaps  a  little  greater.  From 
A  to  B,  the  pressure  falls  considerably.  From  B  to  P  the  pressure  rises. 

Notwithstanding  the  temperature  gradient  from  B  to  A,  the  pressure 
at  A  is  much  greater  than  at  B  because,  although  volume  for  volume  the  air 
at  A  is  considerably  lighter  than  that  at  B,  still  there  is  much  more  of  it  and 
it  rises  to  a  greater  height.  The  path,  therefore,  of  a  particle  in  the  middle 
circulation  is  wholly  independent  of  the  temperature  gradient,  but  depends 
upon  the  gravitational  gradient  or  slope  down  which  it  tends  to  fall  to  the 
pole.  In  other  words,  if  it  were  not  for  the  centrifugal  force,  the  warmer 
column  of  air  at  A  would  prevail  against  the  colder  column  at  B.  Actually 
they  are  balanced  in  a  rather  delicate  equilibrium. 

Let  us  follow  the  course  of  a  particle  just  outside  the  equatorial  circula- 
tion. Since  it  cannot  cross  the  high  pressure  ridge,  which  is  heaped  full,  it 
will  continue  along  parallel  to  this  ridge  as  long  as  the  centrifugal  force  is 
equal  to  the  component  of  gravity  urging  it  down  hill.  The  force  tending 
to  reduce  its  centrifugal  force  is  friction,  and  this  will  be  more  pronounced 
in  the  lower  levels  than  at  the  upper.  Close  to  the  ground  we  know  that 


GENERAL  CIRCULATION  OF  THE  ATMOSPHERE  57 

the  velocity  is  very  much  reduced.  This  reduction  of  velocity  from  friction 
will  cause  it  to  move  towards  the  pole  especially  in  the  lower  levels.  Since 
•the  moment  of  momentum  is  preserved  it  will  soon  acquire  a  velocity  which 
will  hold  it  momentarily  in  equilibrium  in  its  new  position.  The  air  from  above 
sinks  in  order  to  take  the  place  of  the  air  which  has  moved  towards  the 
pole.  This  relieves  the  gravitational  gradient  above  and  the  air  in  the  upper 
levels  which  has  been  pressing  towards  the  equator  takes  the  place  of  the 
air  which  has  sunk.  A  circulation  is  thus  started  which  consists  in  a 
gradual  movement  towards  the  pole  in  the  lower  levels  and  a  movement  out- 
ward in  the  higher  levels.  The  air  moving  poleward  along  the  lower  levels 
displaces  the  air  in  front  of  it  by  raising  it.  In  its  elevated  position  it  is  less 
subject  to  friction  and  quickly  assumes  the  higher  velocities  of  the  upper 
layers.  It  is  now  ready  to  move  outward.  The  movement  is  not  due  to 
its  own  temperature,  but  against  or  in  spite  of  its  own  temperature  gradient. 
The  driving  force  is  derived  from  the  heat  or  temperature  gradient  of  the 
equatorial  circulation.  The  increase  of  velocity  of  a  particle  in  the  higher 
levels  at  a  high  latitude  as  it  moves  along  this  level  to  a  lower  latitude  is 
effected  by  the  friction  of  the  equatorial  circulation  on  the  equatorial  border 
of  the  middle  circulation. 

It  is  not  to  be  supposed  that  there  is  a  definite  circulation  here  by  which 
a  particle  sinks  near  the  lower  border  and,  after  moving  along  the  surface 
to  the  extreme  upper  border,  rises  there  to  make  its  way  along  the  higher 
levels  to  the  lower  border  again.  There  is  not  a  single  simple  circulation, 
but  rather  a  series  of  circulations  all  moving  to  the  east,  only  to  a  slight 
extent  wholly  independent  of  each  other,  and  all  more  or  less  merged  and 
changing  from  time  to  time.  On  the  discs  of  Jupiter  and  Saturn  we  can 
detect  by  the  aid  of  a  telescope  several  such  semi-independent  circulations 
forming  the  middle  circulation,  and  all  undoubtedly  move  to  the  east. 

The  results  of  such  a  circulation  are  that  the  highest  pressure  is  at  the 
lower  border,  and  the  lowest  pressure  at  the  upper  border.  The  surface 
winds  will  in  general  be  somewhat  south  of  west,  at  least  in  the  lower  lati- 
tudes, while  the  upper  currents  will  in  general  be  somewhat  north  of  west. 

With  the  shifting  of  the  circulations  with  the  seasons  slight  changes  may 
be  produced  in  the  direction  of  the  currents  in  the  lower  levels,  so  that  there 
may  be  at  times  W.  and  N.  W.  winds,  but  the  general  trend  is  not  far  from 
west.  Any  one  living  in  temperate  latitudes  with  the  slightest  powers  of 
observation  must  have  noticed  the  general  eastward  direction  of  all  winds. 

Professor  James  Thomson  pointed  out  in  a  general  way  that  the 
prevalence  of  S.  W.  winds  in  the  north  temperate  zone  might  be  explained 
by  the  poleward  pressure  at  the  surface  combined  with  the  differential 
velocity  of  the  earth.* 


*  British  Association  Report,  1857. 


58  THE  ATMOSPHERE 

Lord  Kelvin  has  also  called  attention  to  the  analogy  existing  between 
this  circulation  and  a  cylindrical  vessel  containing  water  which  is  set  in 
rotation  by  stirring.  The  centrifugal  force  heaps  up  the  water  around  the 
edges.  The  friction  on  the  bottom  retarding  the  rotatory  velocity  and  thus 
decreasing  the  centrifugal  force  of  the  particles  on  the  bottom  renders  them 
unable  to  withstand  the  increased  pressure  at  the  edges  due  to  the  heaping 
up  of  the  water  there.  They  are  thus  forced  towards  the  center,  where  on 
rising  to  the  surface  they  quickly  assume  the  velocities  of  their  neighbors 
and  move  outwards  under  the  influence  of  centrifugal  force,  which  is  here 
less  hindered  by  friction.  There  is  thus  a  continuous  circulation,  consisting 
of  a  spiraling  outward  along  the  upper  layers,  a  sinking  at  the  edge  and 
then  a  spiraling  inward  along  the  lower  layers.* 

Ferrel  criticises  this  explanation  as  follows,  in  his  "Motion  of  Fluids 
and  Solids  Relative  to  the  Earth" : 

"At  the  meeting  of  the  British  Association  for  the  Advancement  of 
Science,  in  the  year  1857,  Mr.  Thomson  read  a  short  paper  in  which  he 
explains  this  accumulation  of  the  atmosphere  and  the  consequent  reversion 
(i.e.,  northing)  of  the  lower  strata  of  the  atmosphere  in  middle  latitudes, 
as  arising  from  the  centrifugal  force  of  the  eastward  motion  of  the  atmos- 
phere, and  illustrates  the  effect  of  such  a  force  by  means  of  a  gyrating  vessel 
of  water  in  which  the  surface  water  recedes  from  the  center,  while  at  the 
bottom  there  is  a  flowing  towards  it.  In  this  paper  nothing  is  said  of  the 
influences  of  the  earth's  rotation,  and  if  he  means  that  the  effect  is  produced 
simply  by  the  centrifugal  force  arising  from  the  eastward  motion  of  the 
atmosphere  relative  to  the  earth's  surface,  independent  of  the  earth's  rota- 
tion, the  force  would  not  be  great  enough  to  produce  any  sensible  effect. 
For  examining  the  expression  of  the  force  which  produces  this  effect,  in 
Section  100,  it  is  seen  that  it  depends  upon  the  earth's  rotation,  since 

-TJ,  the  angular  velocity  of  the  atmosphere  relative  to  the  earth,  is  small  in 

comparison  with  2  u,  which  is  double  the  velocity  of  the  earth's  rotation." 
This  criticism  of  Ferrel's  is  just,  for  the  centrifugal  force  which  causes 
the  heaping  up  at  the  border  between  the  equatorial  and  middle  circulations 
is  the  function  R  sin  #  cos  $  (00"  —  ip")  and  not,  as  Kelvin  supposed,  R  sin 
#  cos  #  ^ra,  where  i/»r  is  the  angular  velocity  of  the  circulation  relatively  to 
the  earth. 

With  this  correction,  however,  there  can  be  no  doubt  but  that  Kelvin's 
explanation  of  the  middle  circulation  is  the  correct  one. 

To  recapitulate,  then,  the  middle  circulation  derives  its  velocity  relatively 
to  the  earth  from  the  equatorial  circulation  by  friction  along  the  border. 
This  rotatory  velocity  causes  a  heaping  up  of  the  air  at  the  equatorial  border 

*Lord  Kelvin  (then  Professor  William  Thomson).  Report  of  the  British 
Association  for  the  Advancement  of  Science,  1857. 


GENERAL  CIRCULATION  OF  THE  ATMOSPHERE  59 

with  increased  pressure,  and  this  pressure  forces  the  air  along  the  surface 
towards  the  pole.  The  rotatory  velocity  of  the  lower  layers  being  diminished 
by  friction,  this  poleward  motion  is  less  opposed  by  centrifugal  force.  On 
arriving  at  their  polar  border  the  particles  rise  and  quickly  acquire  the  in- 
creased velocity  of  the  upper  layers.  Centrifugal  force  now  becomes  pre- 
dominant and  drives  them  outward  along  the  upper  levels,  and  thus  the 
circulation  is  established. 

We  have  thus  a  satisfactory  theory  of  the  general  circulation  of  the 
atmosphere  which  agrees  well  with  observed  facts.  Some  of  the  deductions 
have  not  yet  been  verified  because  our  knowledge  of  the  motions  of  the  upper 
atmosphere  is  so  incomplete.  The  knowledge  we  have  derived  from  the 
smoke  of  volcanoes  and  sounding  observations  agrees  with  what  we  should 
expect. 

PLANETARY  CIRCULATIONS 

The  proposition  would  seem  to  be  established  that  an  atmosphere  on  a 
rotating  planet,  having  its  maximum  surface  temperature  at  the  equator  and 
its  minimum  surface  temperature  at  the  pole,  naturally  divides  itself  into  six 
independent  circulations.  The  most  important  are  the  two  equatorial  circula- 
tions which,  by  its  temperature  potential,  carries  on  its  own  as  well  as  the 
middle  circulation.  The  middle  circulations  may  according  to  circumstances 
divide  themselves  up  into  several  more  or  less  differentiated  circulations, 
though  the  borders  between  these  circulations  are  neither  permanent  nor 
marked. 

The  polar  circulations  are  driven  by  their  own  temperature  potentials, 
but  differ  from  the  equatorial  circulations  in  that  while  in  the  latter  deflective 
forces  have  free  play,  in  the  former  they  are  overcome  by  gravitational 
forces.  In  the  middle  circulations  the  temperature  potential  is  overcome  by 
centrifugal  and  gravitational  forces. 

The  width  of  the  equatorial  and  polar  circulations  depends  upon  their 
temperature  potentials  and  the  rotatory  velocity  of  the  planet. 

When  we  have  spoken  of  independent  circulations,  it  is  not  to  be  under- 
stood that  this  is  absolutely  the  case.  There  is,  of  course,  always  more  or 
less  of  interchange  of  particles  between  the  different  circulations,  and,  given 
sufficient  time,  every  particle  will  at  some  time  or  other  occupy  every  possible 
position.  Notwithstanding,  we  can  consider  the  circulations  as  practically 
independent — a  very  fortunate  circumstance,  as  by  dividing  up  the  circula- 
tions the  earth  is  rendered  habitable,  which  would  not  be  the  case  with  a  single 
circulation. 

It  has  been  supposed  that  because  the  weight  of  the  atmosphere  was  less 
at  the  equator  than  at  the  tropics,  there  must  be  a  hollowing  out  or  actual 
deficiency  at  the  upper  surface  there.  On  the  contrary,  there  is  an  actual 
bulging  at  the  equator  at  all  levels,  since  it  is  down  the  slope  of  this  bulge 


6o 


THE  ATMOSPHERE 


that  the  antitrades  slide  to  the  tropics.  A  general  principle  of  some  impor- 
tance regarding  the  atmosphere  as  a  whole  is  the  following.  If  starting 
with  the  atmosphere  at  a  uniform  temperature  and  giving  the  frictional 
forces  time  enough  to  bring  all  particles  to  a  state  of  relative  rest  with  the 
earth,  we  gradually  brought  the  surface  of  the  earth  to  its  actual  tempera- 
tures, then  the  circulations  which  would  be  set  up  would  be  wholly  the  result 
of  temperature  potentials.  These  motive  forces  acting  on  the  whole  along 
meridians,  the  moments  of  those  forces  about  the  axis  of  the  earth  would 
always  be  zero.  Consequently  the  velocity  of  the  atmosphere  as  a  whole 
about  the  axis  of  the  earth  could  not  be  increased  and  the  sum  of  the 
moments  of  the  velocities  of  all  the  particles  would  always  be  equal  to  the 
sum  of  the  moments  of  the  velocities  of  the  earth's  surface  into  the  mass  of 
the  atmosphere. 

Ferrel  expresses  this  in  the  following  manner: 

"It  is  evident,  however,  that  the  east  and  west  motions*  of  the  atmos- 
phere at  the  earth's  surface  multiplied  into  its  distance  from  the  axis  of 
rotation,  must  be  zero;  else  the  velocity  of  the  earth's  rotation  would  be 
continually  accelerated  or  retarded,  which  cannot  arise  from  any  mutual 
action  between  the  surface  of  the  earth  and  the  surrounding  atmosphere." 


ANNUAL  MEAN  OF  PRESSURES  FOR  LATITUDE 


Lat. 

P.  Mms. 

Lat. 

P.  Mms. 

Lat. 

P.  Mms. 

Lat. 

P.  Mms. 

+  80° 

760.5 

40° 

762.0 

0° 

758.0 

40° 

760.5 

75 

760.0 

35 

762.4 

-5 

758.3 

45 

757-3 

70 

758.6 

30 

761.7 

10 

759-1 

5° 

753-2 

65 

758.2 

25 

760.4 

15 

760.2 

55' 

748.2 

60 

758.7 

20 

759-2 

20 

761.7 

60 

743-4 

55 

759-7 

15 

758.3 

25 

763.2 

65 

739-7 

5° 

760.7 

JO 

757-9 

3° 

763-5 

70 

738.0 

45 

761.5 

5 

758.0 

35 

762.4 

*  Meaning  velocities  relatively  to  the  earth. 


CYCLONES 


WE  shall  next  take  up  the  subject  of  abnormal  disturbances  of  the 
atmosphere,  such  as  cyclones,  tornadoes,  thunderstorms,  etc.  Besides  their 
scientific  interest,  these  phenomena  possess  for  us  an  intensely  human 
interest.  For  this  reason  a  digression  of  a  purely  descriptive  character  may 
be  permitted  before  we  take  up  their  formal  mathematics,  especially  since 
comparatively  few  persons,  including  meteorologists,  have  ever  witnessed 
or  ever  will  witness  a  tropical  hurricane. 

The  following  description  of  a  West  Indian  hurricane  by  Alexander 
Hamilton  is  interesting  from  its  accuracy  and  its  quaintness.  It  was  the 
first  writing  of  this  distinguished  man  ever  published.  He  was  at  the  time 
quite  a  young  man,  visiting  friends  in  St.  Croix,  D.  W.  I.  On  August  31, 
1772,  a  hurricane  of  small  diameter  but  considerable  violence  passed  over 
St.  Croix,  and  the  following  description,  which  Hamilton  sent  to  his  father, 
was  published  in  the  local  paper.  It  shows  the  effect  produced  upon  a  bright 
young  mind  by  one  of  these  terrific  disturbances. 

"Sx.  CROIX,  September  6,  1772. 

"HONOURED  SIR:  I  take  up  my  pen  just  to  give  you  an  imperfect 
account  of  the  most  dreadful  hurricane  that  memory  or  any  records  whatever 
can  trace,  which  happened  here  on  the  3ist  ultimo  at  night. 

"It  began  about  dusk,  at  north,  and  raged  very  violently  till  ten  o'clock. 
Then  ensued  a  sudden  and  unexpected  interval,  which  lasted  about  an  hour.* 
Meanwhile,  the  wind  was  shifting  round  to  the  southwest  point,  from 
whence  it  returned  with  redoubled  fury  and  continued  so  till  near  three 
o'clock  in  the  morning.  Good  God !  What  horror  and  destruction — it's 
impossible  for  me  to-  describe,  or  you  to  form  any  idea  of  it.  It  seemed 
as  if  a  total  dissolution  of  nature  was  taking  place.  The  roaring  of  the 
sea  and  wind — fiery  meteors  falling  about  in  the  air — the  prodigious  glare  of 
almost  perpetual  lightning — the  crash  of  the  falling  houses,  and  the  ear-pierc- 
ing shrieks  of  the  distressed  were  sufficient  to  strike  astonishment  into  angels. 
A  great  part  of  the  buildings  throughout  the  island  are  leveled  to  the  ground 
— almost  all  the  rest  very  much  shattered — several  persons  killed  and  num- 
bers utterly  ruined — whole  families  running  about  the  streets  unknowing 
where  to  find  a  place  of  shelter — the  sick  exposed  to  the  keenness  of  water 
and  air — without  a  bed  to  lie  upon — or  a  dry  covering  to  their  bodies — our 
harbour  is  entirely  bare.  In  a  word,  misery  in  all  its  most  hideous  shapes 

*  The  cyclone  center  passed  directly  over  St.  Croix. 


62  THE  ATMOSPHERE 

spread  over  the  whole  face  of  the  country.  A  strong  smell  of  gunpowder 
added  somewhat  to  the  terrors  of  the  night;  and  it  was  observed  that  the 
rain  was  surprisingly  salt.  Indeed,  the  water  is  so  brackish  and  full  of 
sulphur  that  there  is  hardly  any  drinking  it.  My  reflections  and  feelings 
on  this  frightful  and  melancholy  occasion  are  set  forth  in  the  following 
self-discourse. 

"Where  now,  oh !  vile  worm,  is  all  thy  boasted  fortitude  and  resolu- 
tion? What  is  become  of  thy  arrogance  and  self-sufficiency?  Why  dost 
thou  tremble  and  stand  aghast?  How  humble,  how  helpless,  how  con- 
temptible you  now  appear.  And  for  why?  The  jarring  of  the  elements — 
the  discord  of  the  clouds  ?  Oh,  impotent  presumptuous  fool !  How  darest 
thou  offend  that  Omnipotence  whose  nod  alone  were  sufficient  to  quell  the 
destruction  that  hovers  over  thee,  or  crush  thee  into  atoms?  .  .  .  Let  the 
earth  rend,  let  the  planets  forsake  their  course,  let  the  sun  be  extinguished 
and  the  heavens  burst  asunder — yet  what  have  I  to  dread?  My  staff  can 
never  be  broken — in  Omnipotence  I  trust.  .  .  .  Hark !  Ruin  and  confusion 
on  every  side.  'Tis  thy  turn  next :  but  one  short  moment — even  now — Oh ! 
Lord,  help — Jesus,  be  merciful !  Thus  did  I  reflect  and  thus  at  every  gust 
of  the  wind  did  I  conclude — till  it  pleased  the  Almighty  to  allay  it.  ... 

"I  am  afraid,  sir,  you  will  think  this  description  more  the  effect  of 
imagination  than  a  true  picture  of  realities.  But  I  can  affirm  with  the  great- 
est truth  that  there  is  not  a  single  circumstance  touched  upon  which  I  have 
not  absolutely  been  an  eye-witness  to." 

As  another  example  of  the  fearful  human  interest  attaching  to  a  tropical 
cyclone,  we  give  a  description  of  the  hurricane  of  August  10,  1856,  which 
devastated  Last  Island  on  the  Louisiana  coast.  Forty-four  years  later 
(September  6,  1900)  a  similar  cyclone  passed  some  miles  farther  to  the  west 
and  devastated  Galveston  with  a  much  greater  loss  of  life.  The  description 
is  by  that  incomparable  word-painter  Lafcadio  Hearn,  who  must  have  wit- 
nessed such  a  cyclone  to  have  described  it  so  accurately. 

"Then  one  great  noon,  when  the  blue  abyss  of  day  seemed  to  yawn  over 
the  world  more  deeply  than  ever  before,  a  sudden  change  touched  the  quick- 
silver smoothness  of  the  waters — the  swaying  shadow  of  a  vast  motion. 
First  the  whole  sea-circle  appeared  to  rise  up  bodily  at  the  sky ;  the  horizon- 
curve  lifted  to  a  straight  line ;  the  line  darkened  and  approached — a  mon- 
strous wrinkle,  an  immeasureable  fold  of  green  water,  moving  swift  as  a 
cloud  shadow  pursued  by  sunlight. 

"But  it  had  looked  formidable  only  by  startling  contrast  with  the  pre- 
vious placidity  of  the  open :  it  was  scarcely  two  feet  high ;  it  curled  slowly 
as  it  neared  the  beach,  and  combed  itself  out  in  sheets  of  woolly  foam  with  a 
low,  rich  roll  of  whispered  thunder.* 

"Swift  in  pursuit  another  followed — a  third — a  feebler  fourth:  then  the 


*  The  advance  waves  running  before  the  cyclone. 


CYCLONES  63 

sea  only  swayed  a  little  and  stilled  again.  Minutes  passed  and  the  im- 
measurable heaving  recommenced — one,  two,  three,  four — seven  long  swells 
this  time; — and  the  Gulf  smoothed  itself  once  more.  Irregularly  the 
phenonemon  continued  to  repeat  itself,  each  time  with  heavier  billowing  and 
briefer  intervals  of  quiet — until  at  last  the  whole  sea  grew  restless  and 
shifted  color  and  flickered  green;  the  swells  became  shorter  and  changed 
form.  Then  from  horizon  to  shore  ran  one  uninterrupted  heaving — one 
vast  green  swarming  of  snaky  shapes,  rolling  in  to  hiss  and  flatten  upon  the 
sand.  Yet  no  single  cirrus-speck  revealed  itself  through  all  the  violet 
heights :  there  was  no  wind ! — you  could  have  fancied  the  sea  had  been 
upheaved  from  beneath. 

"Still  the  sea  swelled  and  a  splendid  surf  made  the  evening  bath  delight- 
ful. Then,  just  at  sundown,  a  beautiful  cloudlike  bridge  grew  up  and 
arched  the  sky  with  a  single  span  of  cottony  pink  vapor,  that  changed  and 
deepened  color  with  the  dying  of  the  iridescent  day.  And  the  cloud  bridge 
approached,  stretched,  strained  and  swung  round  at  last  to  make  way  for 
the  coming  of  the  gale — even  as  the  light  bridges  that  traverse  the  dreamy 
Teche  swing  open  when  the  luggermen  sound  through  their  conch-shells  the 
long,  bellowing  signal  of  approach. 

"Then  the  wind  began  to  blow ;  it  blew  from  the  northeast,  clear,  cool. 
It  blew  in  enormous  sighs,  dying  away  at  regular  intervals,  as  if  pausing  to 
draw  breath.  All  night  it  blew :  and  in  each  pause  could  be  heard  the 
answering  moan  of  the  rising  surf — as  if  the  rhythm  of  the  sea  moulded  itself 
after  the  rhythm  of  the  air — as  if  the  waving  of  the  water  responded  precisely 
to  the  waving  of  the  wind — a  billow  for  every  puff,  a  surge  for  every  sigh. 

"The  waves  were  running  now  at  a  sharp  angle  to  the  shore ;  they  began 
to  carry  fleeces,  an  immeasurable  flock  of  vague  green  shapes,  wind-driven 
to  be  despoiled  of  their  ghostly  wool.  Far  as  the  eye  could  follow  the  line 
of  the  beach,  all  the  slope  was  white  with  the  great  shearing  of  them.  Clouds 
came,  flew  as  in  a  panic  against  the  face  of  the  sun  and  passed.  All  that 
day  and  through  the  night  and  into  the  morning  again  the  breeze  continued 
from  the  northeast,  blowing  like  an  equinoctial  gale. 

"Colossal  breakers  were  herding  in,  like  moving  leviathan-backs,  twice 
the  height  of  a  man.  Still  the  gale  grew,  and  the  billowing  waxed  mightier, 
and  faster  and  faster  overhead  flew  the  tatters  of  the  torn  cloud.  The  gray 
morning  of  the  Qth  wanly  lighted  a  surf  that  appalled  the  best  swimmers; 
the  sea  was  .one  wild  agony  of  foam,  the  gale  was  rending  off  the  heads  of 
the  waves  and  veiling  the  horizon  with  a  fog  of  salt  spray.  Shadowless  and 
gray  the  day  remained ;  there  were  mad  bursts  of  lashing  rain.  Evening 
brought  with  it  a  sinister  apparition,  looming  through  a  cloud  rent  in  the 
west — a  scarlet  sun  in  a  green  sky.  His  sanguine  disc,  enormously  magni- 
fied, seemed  barred  like  the  body  of  a  belted  planet.  A  moment  and  the 
crimson  specter  vanished  and  the  moonless  night  came. 


64  THE  ATMOSPHERE 

"Then  the  wind  grew  weird.  It  ceased  being  a  breath;  it  became  a 
voice  moaning  across  the  world — hooting — uttering  nightmare  sounds — - 
Whoo !  Whoo !  Whoo ! — and  with  each  stupendous  owl-cry  the  moving  of 
the  waters  seemed  to  deepen,  more  and  more  abysmally,  through  all  the 
hours  of  darkness.  From  the  northwest  the  breakers  of  the  bay  began  to 
roll  high  over  the  sandy  slope  into  the  salines ;  the  village  bayou  broadened 
to  a  bellowing  flood.  So  the  tumult  swelled  and  the  turmoil  heightened  until 
morning — a  morning  of  gray  gloom  and  whistling  rain.  Rain  of  bursting1 
clouds  and  rain  of  wind-blown  brine  from  the  great  spuming  agony  of 
the  sea. 

"Cottages  began  to  rock.  Some  slid  away  from  the  solid  props  upon 
which  they  stood.  A  chimney  tumbled.  Shutters  were  wrenched  off; 
verandas  demolished.  Light  roofs  lifted,  dropped  again  and  flapped  into 
ruin.  Trees  bent  their  heads  to  earth.  And  still  the  storm  grew  louder  and 
blacker  with  every  passing  hour. 

"Almost  every  evening  throughout  the  season  there  had  been  dancing  in 
the  great  hall :  there  was  dancing  that  night  also.  The  population  of  the 
hotel  had  been  augmented  by  the  advent  of  families  from  other  parts  of  the 
island,  who  found  their  summer  cottages  insecure  places  of  shelter;  there 
were  nearly  four  hundred  guests  assembled.  Perhaps  it  was  for  this  reason 
that  the  entertainment  had  been  prepared  upon  a  grander  plan  than  usual, 
that  it  assumed  the  form  of  a  fashionable  ball.  And  all  those  pleasure- 
seekers — representing  the  wealth  and  beauty  of  the  Creole  parishes — 
whether  from  Ascension  or  Assumption,  St.  Mary's  or  St.  Landry's,  Iberville 
or  Terrebonne,  whether  inhabitants  of  the  multi-colored  and  many-balconied 
Creole  quarter  of  the  quaint  metropolis,  or  dwelling  in  the  dreamy  paradises 
of  the  Teche — mingled  joyously,  knowing  each  other,  feeling  in  some  sort 
akin — whether  affiliated  by  blood,  connaturalized  by  caste  or  simply  inter- 
associated  by  traditional  sympathies  of  class  sentiment  and  class  interest. 

"Perhaps  in  the  more  than  ordinary  merriment  of  that  evening  some- 
thing of  nervous  exaltation  might  have  been  discovered — something  like  a 
feverish  resolve  to  oppose  apprehension  with  gayety,  to  combat  uneasiness 
by  diversion.  But  the  hours  passed  in  mirthf ulness ;  the  first  general  feel- 
ing of  depression  began  to  weigh  less  and  less  upon  the  guests;  they  had 
found  reason  to  confide  in  the  solidity  of  the  massive  building;  there  were 
no  positive  terrors,  no  outspoken  fears ;  and  the  new  conviction  of  all  had 
found  expression  in  the  words  of  the  host  himself — 'II  n'y  a  rien  de  mieux 
a  faire  que  s'amuser.'  Better  to  seek  solace  in  choregraphic  harmonies,  in 
the  rhythm  of  gracious  motion  and  perfect  melody,  than  hearken  to  the  dis-. 
cords  of  the  wild  orchestra  of  storms,  wiser  to  admire  the  grace  of  Parisian 
toilets,  the  eddying  of  trailing  robes  with  its  fairy-foam  of  lace,  the  ivorine 
loveliness  of  glossy  shoulders  and  jeweled  throats,  the  glimmering  of  satin- 


CYCLONES  65 

slippered  feet — than  to  watch  the  raging  of  the  flood  without,  or  the  flying 
of  the  wrack. 

"Night  wore  on;  still  the  shining  floor  palpitated  to  the  feet  of  the 
dancers ;  still  the  pianoforte  pealed,  and  still  the  violins  sang — and  the  sound 
of  their  singing  shrilled  through  the  darkness,  in  gasps  of  the  gale. 

'  'Waltzing!'  cried  a  sea  captain,  'God  help  them !  God  help  us  all  now ! 
The  wind  waltzes  to-night  with  the  sea  for  his  partner.' 

"O  the  stupendous  Valse-Tourbillon !  O  the  mighty  dancer !  One-two- 
three  !  From  the  northeast  to  east,  from  east  to  southeast,  from  southeast 
to  south ;  then  from  the  south  he  came,  whirling  the  sea  in  his  arms.  Some 
one  shrieked  in  the  midst  of  the  revels ;  some  girl  who  had  found  her  pretty 
slippers  wet.  What  could  it  be?  Then  streams  of  water  were  spreading 
over  the  level  planking,  curling  about  the  feet  of  the  dancers. 

"What  could  it  be?  All  the  land  had  began  to  quake,  even  as  but  a 
moment  before  the  polished  floor  was  trembling  to  the  pressure  of  circling 
steps ;  all  the  building  shook  now ;  every  beam  uttered  its  groan.  What  could, 
it  be?  There  was  a  clamor,  a  panic,  a  rush  to  the  windy  night.  Infinite 
darkness  above  and  beyond ;  but  the  lantern  beams  danced  far  out  over  an 
unbroken  circle  of  heaving  and  swirling  black  water.  Stealthily,  swiftly,  the 
measureless  sea-flood  was  rising.* 

"For  a  moment  there  was  a  ghastly  hush  of  voices.  And  through  that 
hush  there  burst  upon  the  ears  of  all  a  fearful  and  unfamiliar  sound,  with 
volleying  lightnings.  Vastly  and  swiftly,  nearer  and  nearer  it  came — a 
ponderous  and  unbroken  thunder-roll,  terrible  as  the  long  muttering  of  an 
earthquake. 

"The  nearest  mainland — across  mad  Caillon  Bay  to  the  sea  marshes — 
lay  twelve  miles  north;  west,  by  the  gulf,  the  nearest  solid  ground  was 
twenty  miles  distant.  There  were  boats,  yes!  but  the  stoutest  swimmer 
might  never  reach  them  now ! 

"Then  rose  a  frightful  cry — the  hoarse,  hideous,  indescribable  cry  of 
hopeless  fear — the  despairing  animal-cry  man  utters  when  suddenly  brought 
face  to  face  with  Nothingness,  without  preparation,  without  consolation, 
without  possibility  of  respite — Sauve  qui  peut. 

"And  then — then  came,  thundering  through  the  blackness,  the  giant 
swells,  boom  on  boom !  One  crash !  The  huge  frame  building  rocks  like  a 
cradle,  seesaws,  crackles.  Another !  Chandeliers  splinter ;  lights  are  dashed 
out ;  a  sweeping  cataract  hurls  in ;  the  immense  "hall  rises — oscillates — twirls 
as  upon  a  pivot — crepitates — crumbles  into  ruin.  Crash  again !  the  swirling 
wreck  dissolves  into  the  wallowing  of  another  monster  billow;  and  a  hun- 
dred cottages  overturn,  spin  in  eddies,  quiver,  disjoint  and  melt  into  the 
seething. 


*  The  cyclone  wave  carried  along  by  the  center  of  the  cyclone. 


66  THE  ATMOSPHERE 

"So  the  hurricane  passed,  tearing  off  the  heads  of  prodigious  waves,  to 
hurl  them  a  hundred  feet  in  air — heaping  up  the  ocean  against  the  land — 
upturning  the  woods.  Bays  and  passes  were  swollen  to  abysses;  rivers 
regorged ;  the  sea  marshes  were  changed  to  raging  wastes  of  water.  Before 
New  Orleans  the  flood  of  the  mile-broad  Mississippi  rose  six  feet  above 
high  water  mark.  One  hundred  and  ten  miles  away  Donaldsonville 
trembled  at  the  towering  tide  of  the  Lafourche.  Lakes  strove  to  burst  their 
boundaries.  Far  off  river  steamers  tugged  wildly  at  their  cables — shivering 
like  tethered  creatures  that  hear  by  night  the  approaching  howl  of  destroyers. 
Smokestacks  were  hurled  overboard,  pilot  houses  torn  away,  cabins  blown 
to  fragments.  And  over  roaring  Kaimbuck  Pass — over  the  agony  of  Caillou 
Bay — the  billowing  tide  rushed  unresisted  from  the  gulf — tearing  and 
swallowing  the  land  in  its  course — ploughing  out  deep-sea  channels  where 
herds  had  been  grazing  but  a  few  hours  before — rending  islands  in  twain — 
and  ever  bearing  with  it,  through  the  night,  enormous  vortex  of  wreck  and 
wan  drift  of  corpses." 

A  whirl  of  air,  in  the  same  direction  as  the  earth  turns  at  the  point 
where  the  whirl  exists,  is  called  a  cyclone.  A  whirl  in  the  opposite  direction 
is  called  an  anticyclone.  We  have  seen  that  the  tendency  of  a  moving  mass 
on  the  earth's  surface  is  to  turn  towards  the  right  in  the  northern  hemisphere, 
and  to  the  left  in  the  southern  hemisphere.  Projected  upon  the  plane  of  the 
equator  these  directions  are  the  opposite  of  the  earth's  rotation  or  anti- 
cyclonic. 

Before  we  come  to  consider  the  formation  of  a  cyclone,  it  may  be  stated 
that  in  such  whirls  the  pressure  is  always  considerably  less  in  the  center 
than  around  the  edges.  Consequently  there  is  always  a  force  urging  the 
gyrating  stream  towards  the  center.  Motion  directly  towards  the  center  is 
prevented  by  the  deflecting  force  which  turns  it  to  the  right  in  the  northern 
hemisphere.  Hence  in  either  hemisphere  a  local  diminution  of  pressure  must 
result  in  a  cyclonic  whirl.  The  deflective  force  we  have  seen  to  be  approxi- 
mately 2.v  co  sin  $,  where  V  is  the  relative  velocity  and  the  other  symbols  have 
their  usual  significance.  The  deflective  forces  and  the  pressure  gradients 
are  thus  opposed  to  each  other.  If  the  component  of  the  deflective  force 
away  from  the  center  is  equal  to  the  pressure  gradient  towards  the  centre, 
then  a  particle  will  be  urged  neither  inward  nor  outward  and  will  tend  to 
gyrate  in  a  circle  about  the  center  of  low  pressure.  If  the  pressure  gradient 
is  predominant  then  the  particle  will  spiral  inward :  if  the  deflective  force  is 
predominant,  then  the  particle  will  spiral  outward.  In  the  dynamics  of  the 
middle  circulation  of  the  atmosphere,  we  have  had  an  example  of  such  con- 
strained motion,  where  the  deflective  force  continually  urges  the  stream 
outward  towards  the  equator,  while  the  pressure  gradient  opposes  this 
motion.  From  this  point  of  view  the  middle  circulation  is  as  truly  a  cyclone 
as  the  smaller  and  more  violent  whirls  which  occur  at  different  points  of  the 


CYCLONES  67 

earth's  surface.  The  middle  circulation  is,  therefore,  cyclonic,  while  the 
polar  circulation  is  anticyclonic,  and  the  equatorial  circulation  is  neither. 

There  are  two  principal  theories  as  to  the  formation  of  cyclonic  whirls. 
The  first  postulates  a  simple  local  heating  of  some  extent  which  gives  rise 
to  a  convectional  ascending  current.  As  the  bottom  of  an  ascending  current 
is  always  a  locus  of  diminished  pressure  for  reasons  already  given,  we  have 
here  our  center  of  low  pressure  and  the  rest  follows  easily.  The  inward 
pressure  is  at  first  predominant  and  forces  the  currents  to  spiral  inward 
towards  the  center,  which  they  turn  with  a  high  velocity,  and  rising  to  levels 
where  the  inward  pressure  gradient  becomes  weakened,  they  finally  spiral 
outward  at  the  top.  The  whirl  is  cyclonic  throughout  and  not,  as  has  been 
frequently  stated,  anticyclonic  at  the  top.  This  theory  is  rather  plausible 
and  it  is  possible  that  some  cyclones  are  formed  in  this  way.  On  the  whole, 
however,  it  is  far  from  satisfactory. 

Ferrel,  who  was  one  of  its  advocates,  says :  "On  account  of  the  non- 
homogeneity  of  the  earth's  surface,  comprising  hills  and  valleys,  land  and 
water,  and  dry  and  marshy  areas,  all  with  different  radiating  and  absorbing 
powers,  and  also  on  account  of  the  frequent  irregular  and  varying  distribu- 
tion of  clouds,  it  must  often  happen  that  there  are  considerable  local  de- 
partures of  temperature  from  that  of  the  surrounding  parts ;  and  if  it  should 
so  happen,  as  it  frequently  must,  that  this  area  is  of  a  somewhat  circular 
form,  and  the  air  has  a  temperature  higher  than  that  of  the  surrounding 
part  of  the  atmosphere,  then  we  have  the  conditions  required  to  give  rise  to 
a  vertical  circulation,  with  an  ascending  current  in  the  interior,  as  described 
above."  This  is  eloquent  pleading  for  a  rather  weak  case.  Few,  if  any, 
of  the  severer  cyclones  (the  tropical  ones)  originate  on  land,  where  non- 
homogeneity  and  the  greatest  local  differences  of  temperature  are  found. 
Tropical  cyclones  usually  originate  on  the  sea,  just  within  the  polar  borders 
of  the  doldrums,  where  no  great  local  differences  of  temperature  exist. 

While,  undoubtedly,  heat  is  necessary  for  the  continued  existence  of  a 
cyclone,  for  a  cyclone  is  after  all  a  heat  engine,  yet  in  the  first  instance  it 
would  seem  that  they  are  usually  started  by  dynamical  agencies  other  than 
heat.  A  whirl  of  leaves  on  a  gusty  day  does  not  depend  upon  any  local 
difference  of  temperature  at  the  point  of  origin,  but  is  due  to  two  or  more 
currents  meeting  obliquely. 

We  thus  come  to  the  second  theory,  the  dynamical  one,  which  refers  the 
origin  of  cyclones  to  two  opposing  sheets  of  wind  meeting  along  an  oblique 
line.  If  we  spread  some  light  powder  on  a  table  and  blow  on  it  simultane- 
ously with  two  bellows  from  opposite  directions,  but  a  little  oblique  to  each 
other,  the  powder  will  rise  from  the  table  in  a  whirl,  and  with  a  little  care 
we  can  determine  the  direction  of  the  whirl. 

Now  the  doldrums  shift  their  position  with  the  seasons,  moving  north 
and  south.  In  the  Atlantic  the  whole  belt  of  doldrums,  although  of  varying 


68  THE  ATMOSPHERE 

width,  always  remains  to  the  north  of  the  equator,  while  in  the  Pacific  and 
Indian  oceans  it  is  alternately  wholly  to  the  north  or  south  of  the  equator. 
The  equatorial  border  of  the  trades  which  limit  this  belt,  owing  to  the  earth's 
deflective  force,  blow  N.  W.  and  S.  W.  when  on  the  wrong  side  of  the 
equator.  That  is,  a  northeasterly  trade  becomes  deflected  into  a  north- 
westerly trade  on  crossing  the  equator,  while  the  southeasterly  trade  turns 
into  a  southwesterly  one  when  north  of  the  equator. 

While  performing  this  seasonal  shift  north  and  south,  it  frequently  hap- 
pens that  the  rearward  trade  overtakes  the  other  one,  which  has  not  retreated 
fast  enough  to  keep  out  of  its  way.  It  thus  happens  that  two  sheets 
become  obliquely  opposed  to  each  other  and  a  whirl  is  thus  set  up.  Severe 
tropical  cyclones  originate  almost  without  exception  while  this  shift  is  being 
effected.  Thus  the  tropical  cyclones  of  the  northern  hemisphere  begin  about 
July  and  continue  up  to  December,  while  the  belt  of  doldrums  is  making  its 
southern  retreat.  The  equatorial  circulation  is,  during  this  time,  becoming 
accelerated  owing  to  an  increasing  temperature  gradient  between  its  limits. 
The  oncoming  N.  E.  trade  is,  therefore,  more  vigorous  than  the  retreating 
S.  W.  trade  and  frequently  overtakes  it.  The  result  is  a  cyclone  or  a  series 
of  cyclones  along  the  opposing  borders.  Since  the  doldrums  never  cross  the 
equator  in  the  Atlantic,  always  remaining  to  the  north  of  the  equator,  there 
are  no  cyclones  in  the  South  Atlantic  within  the  tropic  zone. 

The  same  thing  takes  place  in  the  southern  hemisphere.  The  belt  of 
doldrums  begins  its  retreat  across  the  equator  in  January  and  continues 
moving  north  until  June.  It  is  within  these  months,  therefore,  that  cyclones 
occur  in  the  South  Pacific  and  South  Indian  oceans.  The  more  vigorous 
S.  E.  trade  overtakes  the  slowly  retreating  N.  W.  trade  and  a  cyclone  is  apt 
to  result  wherever  their  edges  touch.  The  greater  amount  of  land  masses 
about  the  north  Atlantic,  resulting  in  a  higher  temperature  potential  for  the 
equatorial  circulation,  as  well  as  the  fact  that  the  doldrums  do  not  cross  the 
equator,  seems  to  explain  the  greater  severity  of  Atlantic  tropical  cyclones 
as  compared  with  cyclones  in  other  parts  of  the  world. 

It  may  be  asked  why  the  trades  meeting  obliquely  should  always  set 
up  a  cyclonic  whirl.  It  would  seem  as  likely  that  anticyclonic  whirls  should 
result  as  cyclonic.  The  answer  is  that  such  whirls  do  occur,  but  since  the 
deflective  force  and  the  pressure  gradient  are  here  in  the  same  direction, 
they  quickly  collapse  or  shut  up.  In  a  cyclonic  whirl,  however,  these  forces 
are  opposed  and  the  tendency  is  to  open  out  and  continue. 

The  dynamic  origin  of  whirls  is  a  matter  of  every-day  observation.  A 
current  of  air  meeting  with  any  resisting  object  is  deflected  or  reflected  onto 
itself  invariably  setting  up  whirls.  The  wind  whistling  around  a  house,  a 
post,  a  fence  or  any  other  obstacle  continually  gives  rise  to  eddies  which  we 
recognize  when  they  snatch  up  light  objects.  A  current  flowing  over  a  sharp 
edge  is  always  full  of  eddies  on  the  leeward  side.  The  Rocky  Mountains 


CYCLONES 


69 


on  the  North  American  and  the  Andes  on  the  South  American  continent, 
serving  as  barriers  to  half  the  atmosphere  in  the  middle  circulation,  set  up 
whirls  on  the  leeward  side,  as  the  westerlies  sweep  against  and  over  them. 
This  is  especially  the  case  in  winter,  when  the  energy  of  this  circulation, 
owing  to  the  increased  temperature  gradient  of  the  equatorial  circulation,  is 
about  four  times  that  of  summer.  Over  the  middle  latitudes  of  the  North 
American  continent,  whirl  after  whirl  of  cyclonic  character  reels  off  from 
the  comb  of  the  Rockies,  every  winter  storm  being  first  noted  as  a  depres- 
sion forming  just  to  the  leew'ard  of  this  ridge  at  a  latitude  where  the  upper 
northwesterlies  blow  most  fiercely.  Certain  it  is  that  if  the  North  American 
continent  were  flat,  its  weather  would  lose  much  of  its  cyclonic  character  and 
consist  chiefly  of  monotonous  "Brave  West  Winds."  On  weighing  the  two 
theories,  therefore,  we  see  that  there  is  little  to  commend  the  local  heat  theory 
upheld  by  Ferrel  and  others,  while,  on  the  other  hand,  there  is  much  in  sup- 
port of  the  dynamical  origin  of  cyclones. 

We  have  next  to  consider  how  a  cyclone  once  formed  perpetuates  itself. 
By  either  theory  the  same  kind  of  a  circulation  results,  i.e.,  the  air  spirals  in 
towards  the  center  along  the  lower  levels  and  spirals  out  again  at  the  top. 
In  rising  through  the  center  it  expands  approximately  adiabatically  and 
hence  condensation  must  result.  In  a  tropical  hurricane  the  air  is  warm 
and  saturated,  containing  a  maximum  amount  of  water.  The  vapor  con- 
tained in  the  air  is  -fa  of  it  by  volume,  and  -jV  by  weight  for  30°  C.  Since 
we  may  suppose  the  air  to  undergo  a  practically  adiabatic  expansion  on 
rising,  the  temperature  falls  at  a  faster  rate  than  the  normal  atmospheric 
rate.  At  no  great  height  practically  all  of  the  vapor  is  condensed.  The 
amount  of  latent  heat  evolved  by  this  condensation  is  very  great.  We  may 
get  an  idea  of  what  it  is  from  the  following  table. 


LATENT  HEAT  OF  VAPORIZATION  AT  DIFFERENT  TEMPERATURES 


Centigrade 

Calories 

Centigrade 

Calories 

0° 

606 

60° 

565 

10 

600 

70 

558 

20 

593 

80 

551 

3° 

586 

90 

544 

40 

579 

IOO 

537 

5° 

572 

The  condensation  of  the  amount  of  vapor  we  have  considered  would 
give  out  heat  enough  to  raise  an  equal  weight  of  air  2467°  C.,  and  it  would 
raise  thirty-seven  times  its  weight  of  air,  or  the  air  containing  it,  66°  C. 
This  would  expand  it  at  constant  pressure  by  nearly  £. 


7o  THE  ATMOSPHERE 

We  thus  get  a  general  idea  of  the  enormous  heating  effect  of  the  con- 
densation of  the  vapor  and  the  consequent  very  considerable  expansion  of 
the  air  in  the  axis  of  the  cyclone.  Once  started,  therefore,  and  given  plenty 
of  vapor  we  can  easily  see  how  the  cyclone  will  persist  or  increase  while 
liberating  an  enormous  amount  of  energy.  An  ordinary  heat  engine  de- 
stroys coal,  thereby  liberating  energy  and  performing  work.  A  cyclone  is 
equally  a  heat  engine,  only  it  destroys  vapor,  in  this  manner  liberating  an 
enormous  amount  of  energy  and  performing  work. 

We  have  now  started  our  cyclone  and  provided  it  with  means  for  its 
continuation.  It  behooves  us  to  consider  next  what  kind  of  forces  it  brings 
into  play,  and  what  forces  oppose  it. 

If  we  regard  only  the  rotatory  velocities,  neglectingthe  radialvelocities, 
or,  in  other  words,  consider  only  the  components  2  r  q>  of  the  cyclonic  circu- 
lation, where  q>  is  the  angular  velocity  of  a  particle  about  the  center  and  r  its 
distance,  neglecting  the  components  2  r  which  eliminate  themselves,  we  shall 
have  at  any  time  a  certain  definite  rotatory  energy  which  we  can  replace  by  a 
solid  rotating  mass  of  properly  selected  dimensions  and  angular  velocity. 
The  masses  of  air  circulating  around  the  center  of  a  cyclone  represent  gyra- 
tory energy  just  as  much  as  a  rotating  flywheel  or  a  liquid  vortex.  In  the 
case  of  a  gyrating  solid  the  particles  are  held  in  a  fixed  relation  to  each  other 
by  the  forces  of  cohesion.  In  a  fluid  vortex  the  particles  are  continually 
changing  their  relative  position,  but  the  congeries  of  positions  as  a  whole 
does  not  change.  That  is  to  say,  while  in  a  rotating  solid  a  certain  velocity 
at  a  certain  point  is  represented  by  the  same  set  of  particles  in  endless 
sequence,  always  preserving  their  mutual  distances,  in  a  rotating  fluid  we 
have  the  same  velocity  represented  at  the  same  point,  but  by  a  continually 
new  set  of  particles. 

The  dynamical  results  are  the  same  in  either  case.  The  stream  lines 
being  held  in  a  fixed  relation  to  each  other  by  the  forces  we  have  considered, 
the  vortex  possesses  a  certain  property  of  solids,  viz.,  shape,  and  thus  in  a 
certain  manner  imitates  a  solid.  The  principle  that  fluids  in  motion  may 
imitate  solids  has  long  been  recognized  and  is  the  foundation  of  Lord  Kelvin's 
vortex  theory  of  atoms. 

Since  we  have  in  a  cyclone  a  rotating  mass  of  air,  and  this  mass,  at  the 
same  time  that  it  is  rotating  about  its  axis,  is  being  carried  around  the  axis 
of  the  earth,  the  combined  system  represents  a  gyroscope.  For  a  gyroscope 
may  be  defined  as  a  mass  having  simultaneous  rotations  about  two  different 
axes.  Since,  therefore,  a  cyclone  is  a  gyroscope,  we  shall  have  to  consider 
carefully  the  properties  of  such  a  doubly-rotating  body.  A  complete  dis- 
cussion of  the  gyroscope  will  be  found  in  the  Appendix. 

We  shall  define  that  portion  of  a  spherical  shell  cut  out  by  a  small 
circle,  as  a  spherical  cap.  In  the  present  discussion  the  space  occupied  by  a 
cyclone  is  a  spherical  cap.  That  is,  we  shall  consider  this  space  to  be  cir- 


CYCLONES  71 

cular,  although  in  reality  it  is,  as  a  rule,  only  approximately  so.  In  a 
spheroid,  like  the  earth,  we  have  seen  that  the  component  of  gravitation 
along  the  surface  always  urges  a  body  towards  the  pole.  For  a  state  of 
equilibrium  this  must  be  balanced  by  the  component  of  centrifugal  force 
acting  towards  the  equator,  or  the  gravitational  component  is  equal  to 
R  sin  A  cos  0  oo\ 

If  we  suppose  the  earth  at  rest  and  a  spherical  cap  resting  on  its  surface, 
it  would,  if  there  were  no  friction,  slide  towards  the  pole.  If  the  earth  were 
rotating,  a  component  of  centrifugal  force,  acting  at  the  center  of  the  cap, 
would  urge  it  towards  the  equator.  If,  now,  we  set  the  cap  rotating  about 
its  center,  still  other  centrifugal  forces  will  be  developed.  These  are  the 
centrifugal  forces  arising  from  the  rotational  velocities  of  the  polar  and 
equatorial  halves  of  the  cap.  Although  these  velocities  are  in  opposite 
directions,  the  centrifugal  force  of  each  acts  towards  the  equator. 

We  shall  call  these  two  sets  of  centrifugal  forces  the  revolutional  and 
the  rotational  centrifugal  forces. 

It  is  further  evident  that  such  a  rotating  cap,  which  at  the  same  time 
that  it  is  rotating  about  its  center  is  being  carried  around  the  axis  of  the 
earth,  is  a  gyroscope.  In  the  case  of  a  cyclone,  therefore,  there  are  four 
independent  forces  acting  along  a  meridian.  The  component  of  gravity  and 
the  gyroscopic  force  both  urge  it  towards  the  pole,  while  the  two  centrifugal 
forces,  viz.,  the  revolutional  and  the  rotational,  urge  it  towards  the  equator,. 

We  have  seen  that  acting  on  each  particle  is  a  force  R  sin  #  cos  $  (GO*  — 
^"),  which  urges  it  towards  or  away  from  the  pole,  according  as  its  com- 
ponent of  centrifugal  force  is  in  defect  or  excess  of  the  gravitational  com- 
ponent at  the  corresponding  point  of  the  earth's  surface.  The  summation 
of  all  these  forces  for  all  the  particles  of  a  cyclone  must  be  the  equivalent 
of  the  four  meridional  forces  we  have  just  considered  and  must  be  capable 
of  being  resolved  into  these  four  forces. 

The  proof  is  as  follows.  Let  us  first  consider  two  points,  one  on  the 
extreme  northern,  the  other  on  the  extreme  southern  edge  of  the  cyclone. 
If  p  be  the  distance  from  the  center  to  these  points  and  q>  their  angular 
velocity  of  rotation,  the  horizontal  velocity  of  these  points  will  be  respectively 


R  cos  f  #0  +  ^J  fa  —  p  q?  and  R  cos  l$c  —  ^\  4\.  +  P  <P,   where  #„  and 
denote  the  latitude  and  horizontal  angular  velocity  of  the  center. 


If  if;   denote   the   resultant  horizontal  angular  velocity  of    a   point, 
,  which  is  #c.  ±  ~,  we  have  for  the  upper 

=  R  cos  ($c  +    i\fa  —  p<p  =  Rco&6  fa 


and  #  its  latitude,  which  is  #c.  ±  ~,  we  have  for  the  upper  point, 


72  THE  ATMOSPHERE 


or  t/>  = =5 =•= .     For  the  lower  point 

A  cos  T? 


cos      - 


cos 


Now  the  resultant  polar  force  at  any  point  is 

R  sin  #  cos  #  (a?5  —  t//1).     Substituting,  we  have  for  the  upper  point 


COS3     *o  '  -*Rp>>e  COS 


,(0 


and  for  the  lower  point. 


n        -        co,        - 


co'  — 


/?'  cos'  («c  -  I)  ^'  +  2  Jf  ft  ip  ^  cos  («„  -  I)  +  P'  <?>' 


•(*) 


From  (i)  we  have 
=  Rsm 


From  (2  we  have) 

=  ^sin^cosi?  (ft?2  -  ty*)-  2  p  p  sin  fo,  -  ^W«-  ^^  tan^c 

Adding,  we  have 


[sin  («.  +  £)  cos  («.  +^)  +  sin  («.  -  |)  cos  («.  -  | 
+  2pvic  [sin  («0  +  ^)  -  sin  (*.  ~  I 


CYCLONES  73 

Developing  and  reducing, 
R  $  =  R  ((i?2  —  ipc)  2  sin  i9c  cos  $c  cos  ^-  +  4  p  <p  ^c  cos  #c  sin  ^ 


/    \ 

.    (3) 


—  tan2  •&„  tana  - 


The  first  term  in  (3)  represents  the  difference  between  the  component 
of  gravitation  applied  at  the  center  and  the  revolutional  centrifugal  component 
applied  at  the  same  point.  The  second  term  is  the  gyroscopic  force  and  the 
last  term  the  rotational  centrifugal  component.  Taking  a  complete  circle  of 
points  at  the  distance  p  from  the  center  we  see  that  for  the  east  and  west 
points  the  first  term  in  (3)  becomes-/?  (GO"  —  ip*)  2  sm  *c  cos  '*«  while  the 
other  two  terms  vanish.  We  may  write  then  as  an  approximate  average  for 
the  circle, 

2  p 

I  -f-  COS  -7j- 

R  $  —  R  (G?  —  if?*}  2  sin  #c  cos  #c  —  -  +  2  p  q>  ipc  cos  i9c  sin  ^ 

sec'^- 
a    ?  'R  . 


-  tana  r9c  tan2 


Since  the  angle  -^  is  generally  small,  especially  in  tropical  hurricanes 
in  their  early  course,  we  may  consider  cos  ~  and  sec  75  as  practically 
unity,  while  sin  ~  is  nearly  -^.  Consequently,  we  can  write  (4). 

R  *  -  2  R  (tf  -  ^')  sin  tf,  cos  *0  +  2  P^  ^  cos  $c  -  ££-  tan  *e.     (5) 

Or,  dividing  by  the  mass, 
R  $  =  A>  (^2  -  y-c')  sin  ^c  cos  tfc  +  fj*  $.  cos  0e  -  ^  tan  da.      (6) 

We  have  so  far  only  considered  the  rim  of  the  cyclone.  Since  the 
average  value  of  p  is  the  radius  of  gyration,  which  we  shall  denote  by  k, 
we  may  write  for  the  whole  cyclone, 

k*  CD  •  /P  <»a 

R  &  =  R  sin  #c  cos  #c  a?"  —  R  sin  #c  cos  $e  tic*  +  ~-  ibc  cos  #c  ---  ^  tan  #„. 

A.  2  A: 

(7) 


74  THE  ATMOSPHERE 

The  terms  are  now  in  an  easily  recognizable  form.  R  sin  $c  cos  $c  GO* 
is  the  gravitational  component,  the  plus  sign  showing  that  it  acts  towards 
the  pole. 

R  sin  #c  cos  &0  ij>*  is  the  revolutional  centrifugal  component,   the* 

k*  cp    • 
minus  sign  showing  that  it  acts  towards  the  equator.      — =-  1/>C  cos  #c  is 

the  expression  for  the  gyroscopic  force  with  which  we  are  already 
familiar.  (See  Appendix.)  The  plus  sign  shows  that  it  acts  towards  the 

/£"  <z>a 

pole.      — j^  tan  #c  is  the  rotational  centrifugal  component,    the  minus- 
2  R 

sign  showing  that  it  acts  towards  the  equator. 

The  rotational  centrifugal  force  itself,  that  is  not  resolved  along  the 

k*   CD* 

surface,  is ^— — — r-. 

2  R  cos  #c 

K*  cp*  is  the  square  of  the  velocity  of  the  whole  mass  concentrated  in 
a  ring  at  the  distance  k  from  the  center,  while  R  cos  $c  is  the  distance  of 
the  center  from  the  axis  of  the  earth,  thus  showing  that  it  is  a  centrifugal 
force. 

We  thus  see  that  the  deflective  forces  of  all  the  points  of  a  cyclone,, 
when  summed  together,  are  equivalent  to  the  four  forces  we  have  just  dis- 
cussed, acting  at  the  center  of  the  cyclone.  Of  these  four  forces  the  two 
centrifugal  forces  act  towards  the  equator,  while  the  gravitational  and 
gyroscopic  forces  act  towards  the  pole.  When  these  forces  are  not  in 
equilibrium  they  will  urge  the  cyclone  either  north  or  south — generally  north, 
in  the  northern  hemisphere — until  a  position  of  equilibrium  is  found,  and 
there  will  always  be  some  latitude  between  the  equator  and  the  pole  where 
this  equilibrium  is  attained. 

Equation  (6),  which  we  may  write 
R  &  =  A  sin  #  cos  #  (o?a  —  ^f)  +  B  cos  0  &  -  C  tan  #         (8) 

cannot  be  solved  until  we  determine  the  manner  in  which  the  fifth  force,  viz.,. 
friction,  acts  upon  the  cyclone. 

This  force  has  evidently  no  effect  in  moving  the  cyclone  in  a  meridional 
direction,  but  it  exerts  a  moment  tending  to  move  it  about  the  axis  of  the 
earth  in  an  anticyclonic  direction,  i.e.,  from  east  to  west. 

If  we  suppose  an  arm  A  B  pivoted  to  a  table  at  A  and  to  a  disc  at  B, 
its  center,  so  that  the  disc  can  turn  about  this  center  while  lying  on  the  table 
and  the  whole  arrangement  is  capable  of  turning  about  A,  we  have,  after  a 
fashion,  a  frictional  model  of  a  cyclone.  If  the  disc  is  turning  in  a  counter- 
clockwise direction,  the  frictional  forces  act  upon  the  lower  half  in  the 
direction  indicated  by  the  arrow,  or  from  east  to  west,  while  the  frictional 


CYCLONES 


75 


forces  on  the  upper  half  act  in  a  contrary  direction.  For  equilibrium  the 
upper  frictional  forces  must  be  greater  than  the  lower,  since  they  act  upon 
a  shorter  arm. 

We  shall  neglect  the  friction  due  to  the  motion  of  the  cyclone  as  a  whole 
over  the  surface  of  the  earth,  as  this  velocity  is  much  less  than  the  rotational 
velocity.  It  is  probable  that  the  friction  of  the  air 
on  the  earth's  surface  is  a  function  that  increases 
rapidly  with  the  velocity — at  least  as  the  square  of 
the  velocity.  Hence,  the  rotational  friction  arising 
from  a  velocity  of  100  to  150  miles  an  hour  will 
far  outweigh  the  translational  friction  due  to  a 
velocity  of  10  to  20  miles  on  an  average.  Hence 
it  seems  allowable  to  neglect  the  latter  in  com- 
parison with  the  former,  and  an  examination  of 
actual  cyclone  paths  appears  to  bear  this  out. 

If,  now,  we  suppose  a  cyclone  held  at  a  cer- 
tain latitude  by  the  opposing  polar  forces  we  have 
just  considered,  and  which,  in  this  sense,  take  the 
place  of  the  arm  A  B  in  Fig.  20,  we  shall  have 
motion  about  the  point  A  if  the  moments  of  the 
two  opposing  frictional  forces  are  not  equal.  This 
motion  is  almost  always,  at  first,  in  an  anticyclonic 
direction,  since  the  frictional  forces  on  the  lower 
half  act  on  a  longer  arm.  There  is,  however,  a  tendency  for  these  two 
moments  to  become  equal,  and  in  actual  cyclones  this  appears  to  be  effected 
in  a  short  time.  This  is  probably  brought  about  by  a  general  deforma- 
tion of  the  cyclone,  the  flow  of  the  air  in  the  equatorial  half  being  im- 
peded by  the  increased  resistance  and  giving  the  air  in  the  polar  half  an 
opportunity  to  spread  out  over  a  greater  surface,  and  thus  by  an  increased 
frictional  area  to  bring  its  moment  up  to  that  of  the  lower  half.  The  actual 
facts  are  that  a  cyclone  preserving  the  same  latitude  always  moves  with  a 
constant  horizontal  velocity,  and  in  moving  from  a  lower  to  a  higher  latitude 
a  cyclone  always  decreases  its  horizontal  velocity.  No  example  has  ever  been 
found  of  a  cyclone  remaining  on  the  same  parallel  and  varying  its  horizontal 
velocity,  nor  any  example  of  a  cyclone  moving  towards  the  pole  and  not 
diminishing  its  horizontal  velocity.  An  example  of  an  actual  cyclone  moving 
with  a  constant  latitude  is  given  in  Fig.  21,  A.  Under  such  conditions  all 
the  forces  must  be  in  equilibrium,  and  a  cyclone  performing  such  a  motion 
may  be  called  a  Poinsot  cyclone.  For  a  body  under  the  action  of  no  forces 
performs  the  motions  we  have  already  investigated  in  the  Appendix,  viz.,  a 
Poinsot  motion,  where  a  mass  rotates  about  two  different  axes  simultaneously 
under  the  action  of  no  forces. 

Let  us  suppose  that  in  a  cyclone,  in  a  certain  latitude,  the  frictional 


FIG.  20 


76  THE  ATMOSPHERE 

forces  on  the  upper  half,  which  tend  to  move  the  cyclone  in  a  cyclonic 
direction,  are  in  equilibrium  with  the  frictional  forces  on  the  lower  half, 
which  tend  to  move  it  in  an  anticyclonic  direction.  If  now  the  cyclone 
moves  to  a  higher  latitude,  these  frictional  forces  will  no  longer  be  in 
equilibrium,  since  the  anticyclonic  moment  will  preponderate  over  the 
cyclonic  moment.  In  other  words,  the  equilibrium  has  been  disturbed  and 
the  preponderant  frictional  forces  on  the  equatorial  side  will  move  the  cyclone 
from  east  to  west  and  cause  a  reduction  of  the  horizontal  velocity.  We  are 
speaking  of  the  absolute  horizontal  velocity  of  the  cyclone,  not  that  relatively 
to  the  earth. 

We  see,  then,  that  a  cyclone  cannot  move  towards  the  pole  without 
losing  some  of  its  absolute  horizontal  velocity,  while  conversely  it  cannot 
move  towards  the  equator  without  gaining  absolute  horizontal  velocity.  As 
it  moves  towards  the  pole  an  excess  moment  comes  into  play  which  we 
may  consider  the  equivalent  of  a  force  /,  acting  to  the  westward,  into  an 
arm^ctn.  #.  If  we  suppose  this  excess  moment  to  remain  constant  from 
point  to  point  as  the  cyclone  progresses  to  the  north,  then  the  force  /  will  be 
inversely  proportional  to  the  arm  or  as  the  tangent  of  the  latitu'de.  The 
amount  of  the  reduction  of  the  horizontal  velocity  will,  therefore,  be  greater 
the  nearer  the  cyclone  approaches  to  the  pole. 

Once  the  cyclone  has  arrived  at  its  latitude  of  equilibrium,  the  excess 
moment  quickly  vanishes  and  the  motion  becomes  uniform.  Since  this 
•excess  moment  only  exists  temporarily  while  the  cyclone  is  moving  from 
point  to  point  towards  the  pole,  we  may  consider  that  the  reduction  of  the 
horizontal  velocity  is  proportional  to  the  force  f.  Consequently  if  v0  be  the 
proper  horizontal  velocity  for  the  cyclone  in  latitude  #0,  then  in  latitude  #  it 
will  be  v0—  K  tan  #,  or  vh  =  .v0—  A"  tan  3,  or  generally  vh  =  C  —  TTtan  #, 
where  C  and  K  are  some  constants  to  be  determined.  We  shall  call  this  the 
tangent  law  of  the  cyclone. 

We  have  departed  in  the  above  somewhat  from  our  former  strict 
methods.  Such  reasoning  is  far  from  strict,  but  merely  plausible.  Under 
the  circumstances  it  seems  hardly  possible  that  the  problem  of  friction  can 
be  subjected  to  strict  analytical  methods.  The  cyclone  is  continually  alter- 
ing its  shape  and  the  configuration  of  its  stream  lines,  this  altering  and 
readjustment  being  itself  an  indication  of  the  frictional  forces  at  work. 
However,  our  reasoning,  though  only  plausible,  will  be  justified  if  it  leads 
to  some  practical  result.  To  judge  of  this,  we  shall  have  to  apply  it  to  some 
actually  charted  cyclones. 

It  may  happen,  and  frequently  does,  that  where  a  cyclone  is  changing 
its  latitude  rapidly,  the  excess  frictional  moment  in  the  anticyclonic  direction 
does  not  have  time  to  reduce  the  horizontal  velocity  to  the  value  indicated 
by  our  law.  In  such  cases  we  shall  find  a  slight  discrepancy  between  the 


CYCLONES 


77 


calculated  and  observed  values,  ranging  from  I  to  4  miles,  and  always  in 
excess  of  the  calculated  value. 

In  Fig.  21  are  given  three  cyclones  charted  by  Reid  and  Redfield.  It 
will  be  seen  that  cyclone  A  is  executing  a  Poinsot  motion ;  that  is,  it  preserves 
its  latitude  as  well  as  a  constant  horizontal  velocity.  Such  cyclones  are  rare 
in  low  latitudes,  as  the  gyroscopic  forces  are  almost  always  in  excess  of  the 
centrifugal. 

For  cyclone  B,  we  have  the  following  data.  We  shall  use  in  these  cal- 
culations geographical  miles  instead  of  statute  miles  as  heretofore. 

On  the  I3th,  it  was  in  Long.  66.75°  W.  On  the  I4th,  it  was  in  Long. 
73.25°.  The  middle  latitude  was  19°  N.  Since  a  degree  of  longitude  at 
Lat.  19°  contains  56.8  geographical  miles,  and  it  had  traveled  6.5° 
in  24  hours,  its  velocity  relatively  to  the  earth  was  15.5  miles.  Since  the 
velocity  of  the  earth  here  is  852 .  i  miles,  its  absolute  horizontal  velocity  was 
836.6  miles. 

Similarly,  between  the  I7th  and  i8th,  it  had  moved  over  4.25° 
of  longitude  with  an  average  latitude  of  27°.  Since  a  degree  at  this  latitude 
contains  53.5  geographical  miles,  it  had  moved  over  227.6  miles  relatively 
to  the  earth  in  24  hours,  or  its  relative  horizontal  velocity  was  9 . 5  miles.  The 
velocity  of  the  earth  at  this  latitude  is  803.2  miles.  Hence,  the  absolute 
horizontal  velocity  was  793.7  miles.  Substituting  these  values  in  our 
formula,  we  have 

836.6=6"  — #"  -34433 
793-7  =  C  —  K.  50953 

By  elimination  we  find  that  C  =  927.2  and  K  =  262.5. 

From  these  parameters  we  can  construct  the  following  table. 

ABSOLUTE  HORIZONTAL  VELOCITY 


Date 

Observed 

Calculated 

13 

836.6 

14 

820.8 

821.2 

15 

814 

813 

16 

805.7 

804.8 

17 

793-7 

18 

For  the  cyclone  C,  we  find  it  at  Lat.  15°,  Long.  77*  W.,  on  the 
26th  September.  On  the  27th  it  was  in  Lat.  16°  3C/  and  Long.  79°.  The 
average  latitude  between  these  two  positions  is  15°  45'.  A  degree  of  longi- 


THE  ATMOSPHERE 


CYCLONES 


79 


tude  at  this  latitude  contains  57.8  miles.  Hence,  there  was  a  relative  velocity 
of  4.8  miles  and  an  absolute  horizontal  velocity  of  862.6  miles.  From  the 
9th  to  loth  October  it  moved  5.5°  of  longitude  in  an  average  latitude  of  34°. 
Since  a  degree  of  longitude  here  is  49.851  miles,  it  moved  with  a  relative 
velocity  of  n  .4  miles  and  with  an  absolute  horizontal  velocity  of  758.8  miles. 
Substituting  in  our  formula,  we  have 

862.6=  C  —  K  .28203 
758. 8  =  C  —  K. 67451 


Whence  we  find  that  C  =  936 . 5  and  K  =  262 . 2. 
the  following  table. 


We  can  thus  construct 


CYCLONE  C.     HORIZONTAL  VELOCITY 


Latitude 

Observed 

Calculated 

15°  45' 

862.6 

17° 

856.9 

856.4 

17°  30' 

853-4 

853-8 

18° 

851.1 

851.3 

19° 

845 

846.2 

21° 

835-1 

835-9 

22° 

829.4 

830.6 

23°  45' 

820.4 

821.2 

26°  30' 

806.5 

805-7 

28°  30' 

797-3 

794.2 

3i°  30' 

778.8 

776 

32° 

774-8 

772.7 

33° 

766.8 

766.1 

34° 

758-8 

It  will  be  noted  that  the  actual  values  slightly  exceed  the  calculated  ones 
where  the  cyclone  was  running  over  the  land. 

We  shall  next  consider  the  Porto  Rican  hurricane  of  August,  1899. 
On  the  7th  it  went  from  Long.  60°  30'  W.  to  Long.  66°  W.  in  the  average 
latitude  16°  50'.  On  the  8th  it  went  from  Long.  66°  to  Long.  70°  30'  in  an 
average  latitude  18°  45'.  Since  a  degree  of  longitude  is  57.5  miles  at  Lat. 
16°  50',  its  relative  velocity  was  13.1  miles,  and  its  absolute  horizontal 
-velocity  851.3  miles.  Since  a  degree  of  longitude,  at  18°  45',  is  56.9  miles, 
its  relative  velocity  here  was  10.7  miles  and  its  absolute  horizontal  velocity 
842 . 6  miles. 


8o 


THE  ATMOSPHERE 


Substituting  in  our  formula, 


851.3  —  C  —  K  .30255 
842. 6  =  C  —  K  .33945 

Whence  we  find  that  C  =923  and  K~=  236.  To  find  the  point  of  recurva- 
tion  we  must  solve  the  equation  ve  =  900  cos  #  =  923  —  236  tan  #,  where 
ve  represents  the  velocity  of  the  earth.  That  is  to  say,  the  point  of  recurva- 
tion  will  be  where  the  earth  and  the  cyclone  are  moving  with  the  same 
velocity.  Lat.  28°  gives  a  velocity  of  797  miles  for  the  cyclone,  and  this  is 
about  the  velocity  of  the  earth  at  this  point.  Consequently  the  path  will 
recurve  near  the  parallel  of  28°. 

It  must  be  remembered  that  the  data  derived  from  the  chart  are  not 
strictly  accurate  and  small  errors,  where  the  two  positions  from  which  the 
constants  are  calculated  are  near  together,  will  make  appreciable  errors  in 
these  constants.  Taking  the  more  probable  values  C  =  927  and  K  =  250, 
we  get  the  following  table : 


PORTO  RICAN  HURRICANE  OF  AUGUST  8,  1899 
HORIZONTAL    VELOCITY 


Latitude 

Observed 

Calculated 

16°  50' 

851 

851.4 

i?° 

849 

850.4 

18° 

845-8 

845-8 

19° 

841.4 

841.0 

19°  30' 

839.2 

838.1 

20°  30' 

837.2 

833.8 

21° 

835.1 

831.3 

22°  30' 

826.2 

823.5 

24°  30' 

814.5 

813.5 

26° 

805.8 

805.3 

27° 

799.2 

799-7 

28° 

794 

794.1 

29° 

788 

788.5 

30° 

782 

782.8 

We  see  both  from  the  calculated  and  observed  positions  that  the  cyclone 
did  not  recurve  sharply,  but  remained  practically  on  the  same  meridian  from 
Lat.  26°  to  30*.  That  is,  it  so  happened  that  the  decrease  of  the  horizontal 
velocity  from  friction  nearly  kept  step  with  the  decrease  of  the  earth's 
velocity  from  the  26th  to  the  3O.th  parallel. 


CYCLONES 
Another  hurricane  given  by  Piddington  had  the  following  data. 


81 


Date 

Latitude 

Longitude 

Oct.    II 

25° 

82°  30'  W 

Oct.    12 

3i°  15' 

82°  15'  W 

Oct.  13 

38°  20' 

78°  W 

Oct.  14 

47°  15' 

68°  30'  W 

-  953 
=  295.4 


Taking  the  average  latitudes  we  have 

HORIZONTAL   VELOCITY 


Date 

Latitude 

Observed 

Calculated 

Ilth  to  I2th 

28° 

796 

796 

1  2th  to  1  3th 

34°  45' 

749-7 

748 

1  3th  to  I4th 

42°  45' 

680 

680 

This  cyclone  went  rapidly  to  the  north  and  must  have  possessed  great 
gyroscopic  force  and,  therefore,  great  energy.  The  preceding  cyclones, 
without  multiplying  examples,  show  that  the  tangent  law  fits  the  facts  rather 
closely.  The  law  has  been  applied  in  a  number  of  other  cases  with  the  same 
result.  No  contradiction  has  been  met  with. 

Whether  we  regard  this  law  as  a  plausible  deduction,  or  merely  as  an 
empirical  formula,  in  either  case  it  supplies  us  with  the  necessary  material 
for  the  completion  of  our  calculation.  We  already  have 


=      sn  0  cos 


R  sin  0  cos 


+ 


tan  *.  (8) 


And  the  tangent  law  which  we  may  write, 
vh  =  R  cos  #  ip  =  a  —  b  tan 


a  b 

=  -f,  sec  #  —  ?B  tan  *  sec 
/v  K 


(9) 


Substituting  (9)  in  (8) 


H 


R 


-R 
P'tpb 
~R~ 


K 


tan*  0  — 

K 


tan* 


82  THE  ATMOSPHERE 

Multiplying  by  ft  and  integrating, 

vj  =. cos  2  ft  -\-  2  (a1  —  b*}  log  cos 

—  V  tan8  ft—(4ab—       J?    }  ft  4-  I- — ^ h  &*  Q? \  log  cos 


-Lab-  ^ 
We  can  reduce  this  to 


cos  2  tf  +  4  a  £  tan  #  —  l>*  tan2  ft 


-  Dft  +  £  log  cosft  +  K        (n) 
or  generally, 

vp*  =  —  A  cos  2  *  +  .5  tan  ft—  C  tan"  *  —  Z>  ft  +  £  log  cos  ft  +  ^T,        (12) 
where  the  capitals  are  constants  in  general. 

,  In  order  to  determine  the  constants  in  Equation  (12)  it  would  be 
necessary  to  observe  the  polar  velocities  at  six  points,  although  theoretically 
Equation  (10)  would  only  require  three  points.  These  equations  are  utterly 
unmanageable  in  practice. 

However,  they  may  be  reduced  by  the.  aid  of  series.     Since  a  function 
can  be  expressed  as  a  series  by  the  aid  of  Maclaurin's  theorem,  which  is 

/  (*)  =/  (o)  +  */V)  +  71  /(")  +  ji  />)  etc- 

we  can  develop  the  functions  in  Equation  (12)  into  the  following  forms, 
by  neglecting  powers  of  ft  above  the  third. 

2  ft9 
cos  2  ft  =  i  —  2  ft\  tan  ft  =  ft-\  --  -. 

O    * 

2  ft3  ft* 

tan"  ft  =  —  -.  log  cos  ft  =  --  . 

3  !  2 

Substituting  these  expressions  in  Equation  (12)  we  have, 

Dft-E-^K.     (13) 


Reducing  and  again  letting  capitals  express  constants  in  general   (not 
the  same  as  before),  we  have 

vp*  =  A*-BV+Cft*-K.        (14) 

The  values  of  ft  are,  of  course,  expressed  here  in  radians.  Since  a  radian 
is  57.295°,  Equation  (14)  only  holds  approximately  for  low  latitudes.  When 
•ft  approaches  unity,  the  series  becomes  less  convergent  and  above  unity  the 
series  is  divergent.  Where  we  limit  ourselves  to  the  third  power  of  ft 
it  would  probably  be  unsafe  to  use  the  formula  for  latitudes  above  40°.  If 
the  latitudes  are  small,  not  above  30°,  we  may  with  some  degree  of  approxi- 
mation omit  the  term  containing  the  cube  and  write 

V  =  A  ft  -  B  ft11  -  K.        (15) 


*^     OF    THE 

UNIVERSITY 

Of 


CYCLONES  83 

As  an  example  we  shall  take  the  Porto  Rican  hurricane  which  we  have 
.already  discussed. 

In  Lat.  16°  this  cyclone  had  a  polar  velocity  of  5  miles  an  hour. 

In  Lat.  1 8°  it  had,  as  nearly  as  we  can  make  out,  the  same  polar  velocity. 

In  Lat.  19.25°  the  polar  velocity  was  4.5  miles. 

Frorn  these  three  points  we  can  calculate  the  constants  in  Equation  (15)- 
Whether  we  write  the  latitude  in  radians  or  degrees  is  immaterial,  except 
that  it  gives  us  different  values  for  the  constants.  We  can,  therefore,  write 
the  three  equations,  using  degrees, 

25  =  A  16       —^256       —  K 

2$  =  Ai&       —8324        -K 

20.25  =  A  19.25  —  B  370.56  —  K 

Eliminating  we  find  that 

A  =  —    12.47 
B  =  -         .365 
K  =  —  131.08 

Substituting  the  value  of  these  constants  in  Equation  (15),  we  find 
that  for  latitude  25°,  vp  =  1/47.5,  which  agrees  with  what  was  actually 
observed,  viz.,  a  velocity  somewhat  less  than  seven  miles  an  hour. 

At  35°  the  polar  velocity  was  somewhat  less  than  12  miles  an  hour,  and 
as  far  as  we  can  judge  it  increased  rapidly  as  it  got  farther  north,  though 
no  record  of  its  subsequent  course  is  available.  Our  formula  (15)  does  not 
carry  us  with  any  safety  above  30°. 

Again,  for  cyclone  C,  which  we  have  already  considered,  we  find  that 

A—    28.3 
B=        .56 
#  =  308. 

This  shows  a  polar  velocity  of  i  mile  an  hour  at  16°,  7  miles  an  hour 
at  25°,  and  zero  velocity  at  35°,  and  for  higher  latitudes  the  polar  velocities 
are  imaginary.  Actually  it  ceased  northing  in  about  36°. 

These  results  are,  of  course,  only  approximate.  It  is  not  to  be  expected 
that  we  can  calculate  the  path  of  a  cyclone  with  the  same  exactness  that  we 
do  that  of  a  heavenly  body.  By  the  aid  of  these  formulas,  however,  we  can 
acquire  a  very  good  idea  of  the  general  shape  of  the  path.  We  can  gen- 
erally determine  about  where  it  will  cease  northing  or  where  it  attains  its 
latitude  of  equilibrium,  where  it  will  recurve  and  block  out  roughly  its 
general  path.  By  path  is,  of  course  meant  the  path  of  the  axis.  Since  the 
cyclone  extends  usually  a  hundred  or  more  miles  on  all  sides  of  the  axis, 
this  knowledge,  although  not  exact,  will  be  useful.  In  the  case  of  a  tropical 
hurricane,  therefore,  it  will  be  possible  to  predict  the  weather  a  week  or 


84  THE  ATMOSPHERE 

more  ahead.  Incidentally,  these  results  are  of  value  in  confirming  our 
theory.  They  show  that  the  motions  of  cyclones  are  practically  what  we 
had  reason  to  believe  must  result  from  the  action  of  the  forces  at  work. 

Ferrel  attempted  to  explain  the  northing  of  cyclones  in  the  following 
manner.  He  says  ("Motions  of  Fluids  and  Solids  Relative  to  the  Earth")  : 
"Now  these  deflecting  forces  being  as  the  sine  of  the  latitude,  the  pressure 
on  the  polar  side  towards  the  pole  is  greater  than  that  on  the  other  side 
towards  the  equator,  and  hence  the  cyclone  moves  in  the  direction  of  great- 
est pressure."  So  far  as  the  latitude  alone  is  concerned,  this  is  true;  but  the 
deflective  force  is  2  V  GO  sin  #,  and  depends  also  upon  the  factor  V ,  which 
may  be  greater  in  a  moving  cyclone  on  the  equatorial  side  than  on  the  polar 
side.  According  to  Ferrel's  explanation,  a  cyclone  should  move  continually 
towards  the  pole  until  it  reaches  it,  which  we  know  is  impossible.  The  polar 
forces  urging  a  cyclone  are,  as  we  have  seen,  a  conbination  of  four  forces, 
gyroscopic,  centrifugal  and  gravitational,  which  are  eventually  derived  from 
the  deflective  forces.  These  forces  at  first  usually  urge  the  cyclone  towards 
the  pole,  but  after  a  time  an  equilibrium  occurs  between  these  four  forces, 
after  which  the  cyclone  remains  upon  its  parallel  of  equilibrium,  or  rather 
oscillates  about  this  parallel  of  equilibrium. 

Meteorology  being  in  its  infancy,  it  has  unfortunately  been  the  custom 
to  attempt  to  explain  all  its  phenomena  off  hand,  at  a  time  when  a  lack  of 
knowledge  of  the  forces  at  work  rendered  such  explanations  impossible. 
The  idea  seems  to  have  prevailed  that  some  explanation  was  necessary,  and 
that  any  explanation  carried  with  it  necessarily  some  advantage.  At  times 
a  certain  cause  has  been  given  for  some  occurrence  and  later  on  the  same 
cause  has  been  called  upon  to  accomplish  a  directly  opposite  effect.  We 
shall  shortly  give  such  an  instance.  In  the  movement  of  cyclones,  the  fact 
that  a  rotational  f  rictional  couple  exists,  tending  usually  "to  turn  the  cyclone 
about  the  axis  of  the  earth  in  an  anticyclonic  direction  has  been  ignored,  as 
well  as  the  fact  that  a  cyclone  is  a  gyroscope  and  that  centrifugal  and  gravi- 
tational forces  are  at  work. 

Among  various  causes  which  have  been  assigned  for  the  motions  of 
cyclones  we  may  mention  the  following.  They  have  been  said  to  be  guided 
by  and  to  follow  up  the  Gulf  Stream. 

The  most  cursory  examination  of  cyclone  paths  will  show  that  this  is 
certainly  not  the  case.  They  have  been  thought  to  be  guided  by  the  coast 
line.  No  further  answer  to  this  is  required  than  that  it  is  evidently  not  so.  It 
is  true  that  in  low  latitudes  the  coast  lines  have  a  certain  general  resemblance 
to  cyclone  curves.  It  is  possible  that  in  remote  times,  when  the  constituents  of 
the  earth's  crust  were  gaseous  and  formed  a  part  of  the  atmosphere,  the  cy- 
clones of  that  time  precipitated  cumulatively  their  contents  along  their  paths 
and  thus  gradually  built  up  the  present  shore  configurations.  The  sun  to-day 
is  in  such  a  condition  and  its  atmosphere  contains  vortices  or  cyclones 


CYCLONES  85 

which  move  along  cyclone  paths  towards  the  pole.  What  are  known  as 
sun  spots  are  in  all  probability  cyclones,  and  their  movement  towards  the 
poles  is  to  be  explained  upon  cyclonic  (gyroscopic)  principles.  To  say, 
however,  that  cyclones  are  guided  by  the  present  shore  lines  is  an  inversion 
of  cause  and  effect.  While  it  is  possible  that  the  present  shore  lines  are 
the  result  of  ancient  cyclones,  it  is  certain  that  the  present  cyclone  paths 
are  not  governed  by  these  shore  lines.  It  is  frequently  stated  in  Weather 
Reports  and  books  on  Meteorology  that  cyclones  are  deflected  by  distant 
areas  of  high  pressure.  Here  again  the  most  cursory  examination  will 
convince  us  that  such  is  manifestly  not  the  case. 

Lastly,  the  east-west  motions  of  cyclones  are  said  to  be  due  to  the 
general  circulation  of  the  atmosphere.  Curiously  the  north-south  com- 
ponents of  the  general  circulation  are  supposed  to  have  no  effect  on  the 
motion  of  cyclones.  This  is  Ferrel's  explanation.  According  to  him, 
cyclones  should  move  towards  the  west  while  in  the  tropics  because  there  is 
here  a  westerly  component  in  the  general  circulation.  That  is  to  say,  there 
is  a  westerly  component  in  the  lower  strata — up  to  15,000  ft.  In  the  higher 
strata  there  are  easterly  components,  but  these  must  have  no  effect.  Again, 
after  passing  the  high  pressure  ridge  (Lat.  34°)  the  general  drift  of  the 
atmosphere  is  towards  the  east ;  hence,  cyclones  must  move  towards  the  east 
in  these  latitudes.  True,  at  the  surface  these  easterly  winds  are  slight  or 
wanting.  However,  in  the  upper  strata  this  drift  is  very  pronounced,  and  it 
is  now  the  upper  strata  which  propel  the  cyclone.  Hence,  these  intelligent 
winds  arrange  it  among  themselves  to  guide  the  cyclone  in  its  proper  path. 
During  its  early  course  the  task  is  allotted  to  the  surface  winds,  while  later 
on  the  higher  currents  assume  all  responsibilities.  We  might  go  on  to  inquire 
why  these  winds  blow  a  cyclone  at  times  so  little  to  the  west  and  then  blow 
a  second  one  very  much  farther  to  the  west,  although  only  a  short  time  has 
elapsed  between  their  passages.  But  it  is  needless  to  pursue  the  subject 
further.  A  careful  examination  of  cyclone  paths  will  show  that  their 
motions  are  practically  independent  of  the  general  circulation. 

A  cyclone  is  a  thin  disc  of  some  hundreds,  perhaps  thousands,  of  miles 
in  diameter.  It  grips  the  earth  by  its  frictional  forces  and  presents  only  its 
thin  edge  to  the  general  circulation.  The  general  currents  on  striking  this 
edge  are  immediately  absorbed  into  the  general  whirl  or  they  may  even  in 
the  highest  strata  flow  over  the  cyclone,  but  it  is  evident  that  they  can  have 
little,  if  any,  effect  in  moving  it.  To  this  there  is  probably  one  exception. 
When  the  whirl  is  formed  in  the  upper  strata  and  has  no  connection  with 
the  earth,  it  is  likely  that  it  is  carried  along  with  the  currents  in  which  it 
was  formed.  The  whirls  formed  on  the  leeward  side  of  a  high  mountain 
comb  which  lies  obliquely  to  a  strong  upper  current,  are  in  all  probability 
carried  along  at  first  with  the  current  until  they  finally  take  root  in  the 
ground.  This  would  seem  to  be  the  case  with  the  whirls  which  are  reeled 


86  THE  ATMOSPHERE 

off  from  the  comb  of  the  Rocky-Andes  range,  though  even  here  gyroscopic 
forces  will  affect  their  course.  Again,  in  the  case  of  very  limited  whirls 
formed  in  the  upper  layers,  such  as  tornadoes,  these  local  manifestations  must 
be  carried  along  with  the  general  currents,  since  they  have  no  footing  on  the 
earth,  or  pied  a  terre,  by  which  to  hold  themselves.  But  observation  shows 
that  well-defined  cyclones,  starting  on  the  earth,  such  as  the  heavy  tropical 
hurricanes,  are  practically  uninfluenced  by  the  general  circulation. 

TORNADOES 

Tornadoes  are  cyclones  of  very  limited  extent  formed  at  a  considerable 
height  above  the  surface  of  the  earth.  The  rotation  of  these  whirls  is 
always  cyclonic — counterclockwise  in  the  northern  hemisphere.  The  theory 
that  this  phenomenon  is  caused  by  the  convectional  ascent  of  a  local  column 
of  heated  air  seems  untenable  for  the  reasons  already  given  in  the  discussion 
of  the  formation  of  cyclones.  As  in  the  former  case,  it  seems  more  prob- 
able that  they  are  formed  by  the  opposition  of  two  currents  meeting  obliquely. 
In  other  words,  they  probably  have  a  dynamical  origin  just  as  is  the  case 
with  the  larger  cyclones.  In  the  equatorial  circulation  we  have  seen  that 
currents  in  nearly  opposite  directions  are  often  overlying  each  other.  In 
the  seasonal  shift  of  the  circulations,  north  and  south,  it  may  be  supposed 
that,  during  the  process  of  readjustment,  a  lower  stream  may  be  pushed  up 
against  a  higher  stream  blowing  in  a  contrary  direction  at  some  point.  In 
fact  this  must  be  frequently  the  case.  The  resulting  whirl  may  be  clock- 
wise or  counterclockwise.  If  the  former,  it  will  quickly  close  up,  as  we 
have  seen,  and  its  existence  will  be  of  short  duration.  If  counterclockwise, 
it  may  persist  provided  it  finds  at  hand  the  necessary  fuel,  or  we  may  say, 
ammunition,  in  the  form  of  water  vapor. 

In  the  middle  circulation  the  same  thing  is  liable  to  happen.  The 
streams  of  the  circulation  flowing  to  the  eastward  are  continually  urged  by 
the  deflective  force  to  turn  to  the  right.  They  are  kept  in  their  course  by 
the  pressure  gradient  which  opposes  this  motion  and  urges  them  towards 
the  pole.  When  these  forces  are  in  equilibrium,  they  will  continue  their 
eastward  motion.  But  it  is  easily  conceivable  that  in  the  seasonal  shifting 
of  this  circulation  these  forces  may  at  some  point  suddenly  fall  out  of 
equilibrium.  If  the  deflective  force  should  suddenly  become  greatly  pre- 
ponderant, the  stream  would  in  fact  turn  to  the  right  and  endeavor  to  short- 
circuit  itself  by  turning  in  a  circle.  This,  in  all  probability,  frequently 
happens.  If,  now,  the  returning  edge,  which  is  moving  to  the  westward, 
comes  into  opposition  to  the  regular  easterly  drift  at  some  point,  a  whirl 
will  result.  If,  as  we  have  pointed  out,  this  whirl  should  be  counterclock- 
wise and  found  plenty  of  saturated  or  nearly  saturated  air  at  hand,  then 
a  tornado  would  result.  There  are  some  essential  points  of  difference  be- 
tween these  upper  whirls  when  they  are  formed  in  the  equatorial  and  when 


CYCLONES 


they  are  formed  in  the  middle  circulation.  Though  having  many  points  in 
common  and  essentially  the  same  phenomena  dynamically,  what  is  known 
as  a  tornado  is  essentially  a  phenomenon  of  the  middle  circulation,  while  a 
waterspout  is  more  a  phenomenon  of  the  equatorial  circulation. 

We  must  suppose  that  a  tornado  at  its  upper  surface — its  surface  of 
formation — has  a  considerable  area,  perhaps  many  square  miles.  At  its 
center  its  velocity  is  greatest,  and  hence  its  friction  on  the  next  underlying 
layers  is  here  greatest.  The  whirl  will  propagate  itself  downward  from  the 
center  in  a  peculiar  funnel-shaped  form.  Its  shape  in  fact  will  be  like 
that  of  the  water  flowing  out  of  a  pipe  at  the  bottom  of  a  basin,  for  the 
following  reasons.  The  centrifugal  force  due  to  the  whirl  must  at  every 
point  be  in  equilibrium  with  the  outside  pressure  of  the  general  atmosphere. 
This  pressure  increases  with  the  depth.  Hence,  the  diameter  of  the  whirl 
must  decrease  the  lower  it  gets,  since  the  centrifugal  force  varies  inversely 
as  this  diameter.  More  strictly,  the  moment  of  the  horizontal  velocity  is 
preserved  as  the  center  bores  its  way  downward  to  the  earth ;  whence  the 
centrifugal  force  is  inversely  as  the  cube  of  the  radius. 

If  we  call  the  height  of  a  point  above  the  earth  z,  and  r  the  radius  of 

C  C* 

the  horizontal  section,  we  have  v  =  —  and  the  centrifugal  force  is  —3-. 

Since  the  velocity  would  be  infinite  at  the 
axis  of  the  tornado,  there  cannot  be  any  air 
there.  In  other  words,  a  tornado  is  a  hollow 
tube  with  an  absolute  vacuum  along  its  axis. 
At  any  point  of  this  sharply  delimited  surface, 
the  forces  urging  a  particle  along  a  tangent  to 
the  surface  in  the  plane  of  the  axis,  are 

C* 

—^  sin  (p  —  £•  cos  (p  (i),  where  <p  is  the  in- 
clination of  this  tangent  to  the  axis  of  the 
tornado. 

The  upward  component,  due  to  the  cen- 
trifugal force,  is  generally  greater  than  the 
downward  component  due  to  gravity. 


FIG.  22 


dr  d  z 

Since  sin  (p  =  -j-,  and  cos  (p  =  -j-,  we  can  write  (i) 

(2) 


r3  '   ds 
Multiplying  by  -7—,  we  have 

CL  T 


d  z      d*  s 
~cTs~  ~d~F' 


dz 


dt      g  d  t~~  df   '  dt 


88  THE  ATMOSPHERE 


Integrating,       -  —  --gz  =  $J  +  K        (3) 

Calling  r0  the  value  of  r  at  the  surface  of  the  earth,  and  v  and  v0 
the  values  of  the  velocity  of  a  particle  in  the  direction  of  the  tangent  to 
the  surface  at  any  point  and  at  the  surface  of  the  earth  respectively,  we  have 


This  equation  shows  that  the  work  of  raising  a  particle  to  a  height  z 
against  gravity,  and  of  increasing  its  kinetic  energy  in  a  vertical  plane 

v  *        v*    . 
from  —  to  —  ,  is  at  the  expense  of  the  centrifugal  energy. 


2          2 


If  the  particle  which  is  whirled  up  by  the  tornado  starts  from  rest  at 
the  surface  of  the  earth,  we  can  write  Equation  (4) 


Such  a  particle  as  we  have  imagined,  therefore,  upon  this  surface,  not 
only  moves  about  the  axis,  but  shoots  upwards.  In  other  words,  it  spirals 
upwards  rapidly  about  the  axis,  and  in  fact  the  appearance  of  a  tornado 
or  a  waterspout  is  that  of  a  great  writhing  rope,  the  upward  spiraling 
streams  resembling  its  strands. 

The  axis  of  the  tornado  being  an  absolute  vacuum,  the  only  case  known 
in  nature,  is  cold.  In  direct  contact  with  this  delimiting  surface  is  saturated 
air  which  is  rapidly  condensed.  Hence,  in  direct  contact  with  the  axial 
vacuum  is  a  shell  of  water  or  at  least  water  drops  with  gaps  of  saturated 
air.  This  water  is  whirled  upwards  against  gravity  until  it  is  thrown  out 
at  the  top  at  some  distance  from  the  axis.  The  latent  heat  from  the  con- 
densation supplies  an  additional  ascensional  force  to  the  air  about  the  core 
which  causes  it  to  whirl  upwards  in  a  steeper  spiral  than  the  core  itself. 

If  we  regard  a  tornado  as  having  so  little  thickness  that  it  may  be 
regarded  as  a  shell,  there  must  be  an  equilibrium  at  any  point  between  the 
general  outer  pressure  of  the  atmosphere  and  the  horizontal  centrifugal  force 
of  the  tornado.  There  are  in  reality  two  centrifugal  forces,  one  due  to  the 

C* 
horizontal  rotation,  or  — r  cos  q),  acting  normally  on  the  surface  outwards  and 

the  centrifugal  force  due  to  the  velocity  upwards  in  a  vertical  plane  through 
the  axis,  which  acts  normally  on  the  surface  inwards.  This  vertical  centrif- 
ugal component,  however,  is  small,  especially  in  the  lower  part  of  the 
tornado,  where  the  vertical  radius  of  curvature  is  very  large. 


CYCLONES  89 

Neglecting  this  vertical  component,  therefore,  we  have  practically  the 

C* 

horizontal  centrifugal  component,  —  cos  q>,  balanced  by  the  general  out- 
side pressure  of  the  atmosphere. 

2^ 

Now  the  atmospheric  pressure  is/0  e  1 ,  where  ^is  the  height  above 
the  surface  of  the  earth,  p0  the  pressure  at  the  surface  of  the  earth,  and 
k  is  the  barometrical  coefficient. 

C*  —  z- 

Or  — r-  cos  m=p0e     k.     Since  the  angle  q>  is  small,  at  least  in  the 
r 

lower  part  of  the  tornado,  the  surface  of  the  tornado  in  its  lower  part 

C* 
can   be   approximately   represented  by  the  formula  —  =  p0  e     kt     this 

-equation  representing  a  vertical  section  through  the  axis.  Such  a  curve 
is  shown  in  Fig.  22,  and  represents  very  nearly  the  form  of  a  tornado. 
A  tornado  may  occur  either  over  land  or  water.  When  originating  over  the 
ocean,  they  are  called  waterspouts.  Still,  as  we  have  before  remarked,  there 
seem  to  be  certain  differences  in  these  phenomena,  whether  on  land  or 
water,  according  as  they  orignate  in  the  middle  or  in  the  equatorial 
circulation. 

In  the  polar  circulation  there  are  no  cyclones  or  tornadoes,  owing  to  a 
lack  of  aqueous  vapor  sufficient  for  their  continued  existence.  Continuous 
gales  are  common,  but  these  are  not  local  phenomena,  being  shared  in  more 
or  less  by  the  whole  circulation. 

Occasionally  the  water  in  the  core  of  a  tornado  after  being  carried  up 
is  discharged  en  masse,  resulting  in  a  so-called  cloudburst  which  deluges 
the  areas  upon  which  it  falls.  It  often  happens  that  tornadoes  in  the  middle 
circulation  carry  the  water  surrounding  their  axes  to  a  height  where  the 
temperature  is  low  enough  to  freeze  it.  Hence,  hail  storms  are  a  frequent 
accompaniment  of  tornadoes.  Hail  is  in  general  an  indication  of  a  whirl 
aloft,  though  when  this  whirl  has  not  extended  its  foot  to  the  ground,  it  is 
not  called  a  tornado  and  its  existence  is  otherwise  unperceived  and  unknown. 
The  axis  of  a  whirl  is  not  necessarily  a  straight  vertical  line,  and  in  fact  it 
frequently  is  not.  When  the  upper  part  of  the  axis  bends  over  so  that 
particles  of  water  at  a  distance  from  the  axis  are  carried  alternately  from  a 
region  below  zero  to  one  above  zero,  it  may  happen  that  the  hail  stones 
show  concentric  layers  of  ice  deposition,  increasing  their  diameter  with 
every  whirl  through  the  colder  spaces. 

A  somewhat  analogous  phenomenon,  but  not  a  tornado,  are  the  moving 
columns  of  dust  which  are  sometimes  encountered  on  deserts.  These  are 
probably  started  by  two  opposing  currents  aloft,  as  are  tornadoes.  An 
opening  is  thus  formed  in  the  upper  colder  layers  through  which  the  in- 
tensely heated  air  from  the  bottom  can  rise.  The  great  difference  of 


90  THE  ATMOSPHERE 

temperature  between  the  top  and  bottom  may  thus  suffice  to  prolong  their 
existence  with  a  minimum  amount  of  vapor  condensation.  But  they  wholly 
lack  the  violence  of  tornadoes,  having  little  energy  and  not  lasting  long. 
Further,  there  is  never  a  vacuum  in  the  axis.  They  are  thus  analogous  to, 
but  not  tornadoes.  So,  too,  are  thunderstorms,  which  consist  of  whirls  in 
the  upper  layers,  the  axes  often  oblique,  and  thus  giving  rise  to  showers 
and  hail. 

A  tornado  rarely  lasts  more  than  two  hours,  during  which  time  it  may 
travel  from  30  to  50  miles  or  more.  The  direction  is  invariably  easterly 
in  the  middle  circulation.  The  velocity  near  the  axis  may  be  greater  than 
500  miles  an  hour. 


OTHER    PHENOMENA    OCCURRING   IN    THE 

ATMOSPHERE 


SOUND 

WE  have  seen  that  any  local  change  in  the  atmosphere  creates  a  con- 
dition of  unstable  equilibrium  which  results  in  the  passage  of  the  disturbance 
from  its  origin  to  outlying  points.  We  have  hitherto  dealt  with  disturbances 
on  a  grand  scale.  We  are  now  to  consider  the  results  of  sudden  changes  of 
pressure  on  a  small  scale. 

Since  a  gas  possesses  inertia  any  sudden  thrust  against  it  will  be 
resisted  just  as  much  as  if  it  were  a  solid.  The  reaction  will  be  equal 
to  the  action.  Since  it  is  elastic,  the  gas  will  be  compressed  between  these 
two  forces  acting  in  opposite  directions.  Since  the  gas  is  free  to  move, 
it  will  move  also  in  the  direction  of  the  force.  The  action  of  a  thrust  is, 
therefore,  double.  It  sets  the  gas  in  motion  and  at  the  same  time  com- 
presses it.  Hence,  we  may  divide  the  work  done  on  the  gas  into  two  parts 
— the  work  done  in  compressing  it  and  the  work  done  in  imparting  to  it 
its  kinetic  energy.  This  double  energy  imparted  to  the  gas  will  at  first 
be  limited  to  the  immediate  vicinity  where  the  disturbance  arose,  but  as 
time  goes  on,  the  changed  portions  of 
the  gas  will  shift  their  positions  and 
the  disturbance  travels  along  as  a  wave 
of  compression  or  rarefaction,  or  both. 

Let  us  suppose  the  gas  divided 
.off  into  laminae  of  equal  thickness, 
i,  2,  3,  4,  etc.,  as  represented  in 
the  upper  part  of  Fig.  23.  Let  us 
further  suppose  that  lamina  I  has  been 
thrust  by  a  force  to  the  right  and 
that  the  disturbance  after  a  certain 
interval  has  extended  to  lamina  4,  so 

that  the  bounding  planes  occupy  the  positions  shown  in  the  lower  part  of 
the  figure.  Let  us  suppose  that  at  this  instant  the  velocities  of  the  laminae 
are  z/lt  vv  vt,  etc. 

The  pressure  of  each  lamina  on  its  neighbor  to  the  right  is  partly 
static  and  partly  kinetic.  Thus  the  pressure  of  lamina  3  on  lamina  4 

d  *u 
is  p0  +  dp^  -f  m  -j—\  where  d '/3  is  the  excess  of  the  pressure  of  3  over 

d'    £ 

that  of  4,  and  —r~  is  the  acceleration  of  3  relatively  to  4,  i  p0  being  the 
general  external  pressure. 


FIG.  23 


92  THE  ATMOSPHERE 

If  we  suppose  that  in  an  infinitesimal  interval  of  time,  dt,  the  laminae 
have  moved  ds^  ds^  ds^  etc.,  we  have  as  the  work  of  compression  on 
lamina  4, 


Likewise  the  work  of  compression  of  lamina  2  on  lamina  3  will  be 
\Po  +  dp^  -f-  d  p3  -\-  m  —j-^\  d  s^  and  the  work  of  compression  of  lamina  i 

on  lamina  2  will  be  (p0  -\-  d  p^  -\-  d  p^  +  dpz  +  m  —  r2 

Since  we  can  neglect  differentials  of  the  second  order  in  comparison 
with  differentials  of  the  first  order,  we  can  write  for  the  work  of  com- 
pression on  the  several  laminae  during  the  time  d  t 

\Po  +  m  -j-^-  )  d  s3  compressional  work  on  4. 
\Po  H~  m  ~T^]  a  '  si  compressional  work  on  3. 
\Po  +  m  ~7"f  I  dsl  compressional  work  on  2. 

Now  the  sum  of  these  infinitesimal  works  of  compression  is  the  total 
compressional  energy  of  lamina  i,  for  we  can  suppose  lamina  I  to  be  com- 
pressed by  successive  steps.  Thus  we  can  compress  lamina  4  into  lamina  3, 
then  lamina  3  into  lamina  2,  and  finally  lamina  2  into  lamina  i. 

Hence,  the  total  compressional  energy  of  lamina  I  is  the  sum  of  the 
expressions  we  have  written  above.  Or  it  is 

*\ 

p0  (dSj  +  ds^  +  ds3)  +  m  2  v  dv. 

o 

Now,  dsl  -f-  ds^  +  ds^  is  the  distance  which  lamina  i  has  been  com- 
pressed, or  the  amount  of  its  compression.  Call  this  distance  c  ;  then  the 
total  compressional  energy  of  lamina  i  is 


Hence,  at  any  point  of  a  wave,  no  matter  what  its  form  or  type,  the 
compressional  energy  is  equal  to  the  normal  pressure  p0  into  the  distance 
the  lamina  has  been  compressed  plus  its  kinetic  energy.  In  other  words, 
its  kinetic  energy  is  equal  to  the  work  of  compression  done  upon  it  in  excess 
of  the  work  of  compression  done  by  the  general  external  pressure. 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE  93 

Since  in  a  vibratory  wave,  during  the  passage  of  one  complete  wave, 
a  lamina  has  moved  forwards  and  back  to  the  identical  point  and  condition 
from  which  it  started,  it  follows  that  the  work  done  upon  it  by  the  general 
pressure  is  zero.  Hence,  for  the  complete  wave  the  total  kinetic  energies 
are  equal  to  the  total  compressional  or  potential  energies. 

A  formal  proof  of  this  theorem  is  given  in  Lord  Rayleigh's  "Theory 
of  Sound,"  par.  245,  where  it  is  stated : 

"It  follows  that  in  a  progressive  wave  of  any  type  one-half  of  the 
energy  is  potential  and  one-half  is  kinetic,"  and  again,  what  amounts  to 
the  same  thing,  "the  total  energy  of  the  wave  is  equal  to  the  energy  derived 
from  compressing  its  whole  mass  from  its  minimum  to  its  maximum  density, 
or  to  the  energy  of  the  whole  mass  moving  with  its  maximum  velocity." 
For  a  complete  vibratory  wave  this  is  true.  For  a  unidirectional  wave, 
such  as  the  compressional  wave  we  have  just  studied,  the  statement  does  not 
apply.  In  such  a  case,  as  we  have  seen,  the  total  compressional  energy  is 
equal  to  the  total  kinetic  energy,  plus  the  total  work  of  compression  of  the 
general  external  pressure p0»  Or  the  total  compressional  energy  of  a  uni- 
directional wave  is  equal  to  its  total  kinetic  energy  plus  the  normal  pressure 
into  the  distance  its  extreme  limiting  plane  (on  the  left)  has  moved  during 
the  time.  In  Fig.  23,  it  is  easily  seen  that  the  sum  of  the  compressions  of 
all  the  laminae,  or  2c,  is  equal  to  the  distance  "«,"  which  the  extreme 
limiting  plane  on  the  left  has  been  thrust  by  the  force. 

Let  us  suppose  that  a  thrust  has  moved  the  adjacent  gas  a  distance  a 
and  then  ceases ;  also  that  during  the  time  of  the  thrust  the  disturbance  has 
extended  to  a  distance  /.  After  the  wave  has  passed,  each  particle  has  been 
moved  along  a  distance  a  and  set  down  at  rest  in  its  new  position. 

Let  us  suppose  that  the  average  velocity  of  a  particle  during  this  time 
is  v.  Then  while  the  disturbance  has  moved  a  distance  I,  the  particle  has 

moved  a  distance  a.     Call  V  the  velocity  of  the  wave.     Then  V  =  -  v. 

The  gas  of  the  wave  occupies  at  normal  pressure  a  volume  /.  Its 
total  compressional  energy  is  practically  the  same  as  that  derived  from 
compressing  a  volume  /  at  normal  pressure  to  a  volume  /  —  a.  By  thermo- 
dynamics, the  work  of  performing  such  a  compression  adiabatically  is 


k- 


fc-i   -. 


a,  the  amplitude  of  the  wave,  is  generally  small,  so  that  we  can  neglect 
higher  powers.  Expanding  the  above  expression  and  neglecting  powers  of 
a  above  the  square,  we  have 


w- 


•h 


k  is  the  ratio  of  the  two  specific  heats. 


94  THE  ATMOSPHERE 

The  kinetic  energy  of  the  wave  is z/2  =  —  •  — — — ,  where  D  is  the 

2  2  / 

absolute  density,  or  mass  per  unit  volume  of  the  medium. 

Now  we  know  from  our  preceding  theorem  that  the  total  compressional 
energy  is  equal  to  the  total  kinetic  energy  plus  the  expression  p0  a. 

Hence,  ,    „  _ 

2i  21 

This  is  the  velocity  of  propagation  of  the  wave  for  amplitudes  so  small 
that  higher  powers  than  the  square  of  the  amplitude  can  be  neglected. 
For  a  negative  or  expansional  wave,  that  is,  a  wave  arising  from  a  thrust 
to  the  left,  it  is  easily  seen  that  the  velocity  is  the  same,  provided  we  neglect 
higher  powers  of  the  amplitude  than  the  square. 

For  a  compressional  or  positive  wave,  if  we  consider  the  third  power 
of  the  amplitude,  we  have 


Hence,  the  velocity  of  a  compressional  wave  increases  with  the  amplitude, 
and  also  with  the  pitch,  or  as  a  increases  or  /  decreases,  since  the  shorter 
the  wave  length  /_,  the  greater  the  number  of  pulses  passing  a  given  point 
in  a  given  time. 

For  a  negative  or  expansional  wave, 


Hence,  a  negative  wave  travels  more  slowly  than  a  positive  wave,  when  the 
amplitude  becomes  appreciable,  and  the  difference  between  their  velocities 
increases  with  an  increasing  amplitude.  This  is  further  self-evident,  since 
a  positive  wave  is  heated  and  a  negative  wave  is  cooled.  It  follows,  there- 
fore, that  the  velocity  of  a  negative  wave  decreases  with  the  amplitude  and 
also  with  the  pitch. 


B 

FIG.  24 


These  results  have  most  important  bearings.  Since  a  vibratory  wave 
is  composed  of  a  positive  and  a  negative  half,  of  equal  amplitudes  and  fol- 
lowing each  other  at  equal  intervals  of  time,  it  will  be  seen  that  the  positive 
half  will  always  tend  to  overtake  the  negative  half  in  front  of  it,  so  that 
the  wave  will  assume  the  form 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE 


95 


and  cannot  be  a  sine  wave,  as  is  usually  assumed.  Sine  waves,  in  fact,  do 
not  exist  in  nature.  The  crests  tilt  forwards,  and  the  hollows  tilt  backwards 
as  shown  in  the  figure.  This  form  of  the  wave  has  most  important  bear- 
ings on  attractional  results,  as  will  be  shown  later. 

As  the  amplitude  increases,  the  difference  between  the  velocities  of  the 
crests  and  hollows  increases,  until  at  length  a  point  is  reached  where  the 
crests  break  over  into  the  hollows,  precisely  as  water  waves  rolling  up  on  a 
beach  break.  Mathematically  expressed,  discontinuities  arise.  The  subject 
of  discontinuities  in  wave  motion  has  been  treated  at  length  by  Riemann, 
Christoffel,  Hugoniot,  Hadamard,  Lamb  and  others. 

Practically,  the  result  is  that  sustained,  rhythmical  •  sounds  of  great 
intensity  are  impossible.  When  a  tuning  fork  or  a  bell  vibrates  excessively, 
,it  loses  the  purity  of  its  tone  as  well  as  rising  in  pitch.  The  waves  break 
and  become  mixed  up,  losing  more  or  less  their  rhythmical  character. 
Ignorance  of  this  principle  has  resulted  in  many  attempts  to  construct 
powerful  musical  instruments  for  the  purpose  of  producing  very  loud  musi- 
cal sounds  which  should  carry  to  great  distances.  Organs  blown  by  steam 
are  an  example.  The  result  has  invariably  been  a  failure.  The  fabled 
music  of  the  spheres  must  have  been  of  rather  small  amplitude. 

Where  a  vibratory  wave  has  considerable  amplitude  and  yet  does  not 
break,  a  compromise  velocity  is  effected  between  the  two  halves,  which  is 


less  than  the  standard  velocity 


/~p 
,  A/  *-j 


of  a  small  amplitude. 


r 


To  understand  the  intimate  mechanism  of  a  vibratory  wave  let  us 
examine  the  accompanying  Fig.  25.  Let  the  line  A  represent  a  row  of  equi- 
distant particles  at  rest.  Then  as  a  wave  train  sweeps  through  them,  their 
.positions  will  become  changed  and  at  a  certain  instant  will  be  as  shown  in 
the  line  B.  The  wave  is  going  in  the  direction  of  the  large  arrow.  The 


96  THE  ATMOSPHERE 

positions  of  a  tuning  fork  as  it  swings  are  shown  under  the  corresponding 
points  of  the  wave.  The  section  e  will  have  been  moved  into  the  position  e'. 
It  is  now  farthest  to  the  right  of  its  position  of  equilibrium,  i.e.,  its  original 
position,  and  is  momentarily  at  rest,  and,  therefore,  of  normal  density. 
The  sections  following  from  right  to  left  indicate  what  its  state  will  be  at 
successive  instants.  It  is  now  swinging  backwards,  i.e.,  against  the  wave, 
and  on  arriving  at  d'  is  at  its  original  position,  but  with  a  maximum  expan- 
sion and,  therefore,  moving  with  a  maximum  velocity.  At  c'  it  is  farthest 
to  the  left,  having  completed  its  excursion  and  come  momentarily  to  rest. 
It  now  swings  forwards,  with  the  wave  to  the  right,  and  on  arriving  at  its 
original  position  at  b'  has  a  maximum  density  and  velocity.  At  a'  it  has 
completed  its  excursion  to  the  right  and  is  at  rest,  with  normal  density,  pre- 
paratory to  executing  its  backward  swing  again.  The  wave  curve  indicates 
the  pressures  or  profile  of  the  wave. 

The  velocity  of  a  compressional  wave  we  have  seen  is 


77  _ 

Z 

"TT" 

while  that  of  an  expansional  wave  is 


2) 


l~p~k 
The  Laplacian  value  for  the  velocity,  \l  *-je-t  is  nearly  332.4  meters- 

per  second  at  o°  C.  This  is  for  very  small  amplitudes.  We  see  that  for 
large  amplitudes  the  velocity  may  become  very  much  greater  than  this. 
Thus,  in  firing  cannon,  if  the  experiment  be  suitably  performed,  the  com- 
mand "Fire"  will  be  heard  after  the  report.  The  heavy  amplitude  of  the 
gun  easily  outstrips  the  weak  energy  of  the  voice. 

Standing  in  the  line  of  fire  of  a  saluting  six-pounder  (smokeless 
powder),  the  author  has,  by  some  rough  determinations,  convinced  himself 
that  the  velocity  of  the  report  was  considerably  greater  than  the  Laplacian 
value. 

Vieille  (R.  Ac.  Sc.  1898-99.  Memorial  des  Poudres  et  Salpetres, 
tome  10,  pp.  177-260)  has  found  in  his  experiments  with  explosive  waves 
in  long  tubes  of  small  caliber,  velocities  up  to  1200  meters  per  second,  and 
greater  ;  which  is  about  four  times  the  Laplacian  value,  or  the  value  usually 
given  as  the  velocity  of  sound. 


OTHER   PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE  97 

Within  the  bore  of  firearms — rifles  and  cannon — the  compressional 
wave  is  transmitted  from  the  breech  to  the  muzzle  with  enormous  velocities. 
On  emerging  from  the  muzzle  into  free  space,  the  wave  undergoes  modifica- 
tions which  influence  its  velocity  materially.  The  wave  is  launched  out  in 
a  certain  sense  as  a  gaseous  projectile.  It  immediately  spreads  out  laterally 
and  at  the  same  time  draws  the  air  towards  itself  from  the  rear  and  sides. 
A  wave  of  rarefaction,  therefore,  follows  close  behind  the  wave  of  com- 
pression. Further,  the  outgoing  lateral  wave  of  compression  touches  along 
a  surface  the  incoming  lateral  wave  of  rarefaction,  and  along  this  surface 
circular  vortices  are  formed.  With  black  powder  this  smoke  ring,  exactly 
like  the  circular  vortices  blown  by  tobacco  smokers,  is  usually  observed 
attending  the  discharge  of  cannon. 

We  have  seen  that  directly  in  front  of  the  gun  the  velocity  of  the  report 
is  greater  than  the  Laplacian  value.  Directly  behind  the  gun  it  is  less,  since 
the  sound  heard  here  arises  from  a  negative  wave.  The  difference  between 
the  times  when  a  report  is  heard  from  a  saluting  ship  as  the  guns  are  fired 
towards  and  away 'from  the  observer,  is  very  appreciable. 

That  an  explosive  wave  of  compression  is  followed  by  a  negative  wave, 
is  rendered  clear  from  the  accompanying  autographs  of  a  small  rapid-fire 
gun  taken  from  MacKendrick's  ".Visible  Speech."  (Fig.  26.) 

For  a  single  explosion,  the  wave  consists  of  a  marked  compressed  or 
positive  portion,  followed  by  a  negative  wave  of  lesser  amplitude. 


MERGED   EXPLOSIONS 
FIG.    26 

A  positive  wave  of  great  amplitude,  traveling  in  free  space,  has  a 
tendency  to  lengthen  and  thus  to  distribute  its  energy  over  a  greater  space 
from  front  to  rear,  in  addition  to  its  inevitable  geometric  distribution  which 
is  inversely  as  the  square  of  the  distance  traveled.  The  swifter  moving 
more  condensed  parts  crowd  up  towards  the  front,  which  thus  becomes 
steep,  and  gives  rise  to  an  ever  increasing  distance  between  the  front  and 
rear.  Thus  gun-fire,  which  is  heard  as  a  short,  sharp  report  near  by,  at  a 
distance  becomes  a  long,  low  rumble.* 

We  have  hitherto  supposed  the  compressions  and  rarefactions  to  be 
effected  adiabatically.  These  may  under  certain  circumstances  take  place 


*  An  analogous  phenomenon  may  be  seen  in  a  high  waterfall.  A  mass  of  water 
may  detach  itself  from  the  main  body,  when  the  lower  part,  which  has  a  slight  start 
over  the  upper  part,  continually  increases  the  distance  between  them,  until  finally  the 
mass  breaks  up  into  drops. 


98  THE  ATMOSPHERE 

nearly  isothermally.  If  c  denotes  the  compression,  the  total  work  of  com- 
pression, if  it  takes  place  isothermally,  is/0  /  /  -. ,  since/  =  j± — . 

J     b  ~~  C  /  ~"~™  C 

Hence,  W  —  p0  /log  ( — j—  \  =  p0  /log  (i  —  -A.  Developing  the  loga- 
rithm by  Maclaurin's  theorem  and  neglecting  powers  of  c  above  the 
square,  since  c  is  small,  W=69l\T\ 75)  =  p0  c -\-  ^-j-.  The  last  term, 

\l          2  I  /  2  / 

p0  c*      I  D  v* 

we   have   seen,    is   equal  to  the  total    kinetic  energy,   or  - — _  —  — 

2  /  2 

D  V*  c* 
=  —  — - — ,  where  v  and  V  denote  the  same  quantities  as  before.    Whence 

2   / 

V '=  \/ /=?•     This  is  the  value  which  Newton  found  for  the  velocity  of 

sound  on  the  assumption  that  the  compressions  and  rarefactions  were 
isothermal.  Now,  a  single  compressional  wave  must  inevitably  give  up  some 
of  its  heat  of  compression.  Further,  when  the  amplitude  is  very  small  and 
the  wave  of  .considerable  length,  the  distinction  between  adiabatic  and 
isothermal  compressions  tends  to  vanish.  For  short  vibratory  waves,  no 
matter  how  small  the  amplitude,  the  compressions  are,  no  doubt,  practically 
adiabatic,  but,  as  we  have  said,  for  long  unidirectional  waves  of  small 
amplitude,  they  may  approach  isothermal  conditions.  We  have  an  example 
of  this  in  an  experiment  which  was  performed  upon  a  grand  scale.  About 
10  A.M.,  local  time,  of  August  27,  1883,  the  volcano  of  Krakatoa  blew  off 
its  top  in  a  culminating  explosion.  This  started  a  compressional  wave  which 
circled  round  the  globe  no  less  than  seven  recorded  times.  The  records 
appeared  as  notches  in  the  tracings  of  the  self-registering  barometers  of  all 
the  meteorological  stations  of  the  world. 

For  29  stations  the  mean  velocity  from  Krakatoa  to  them  on  the  first 
lap  was  707  English  miles  per  hour.  For  27  stations,  from  the  first  to  the 
third  passage,  or  after  the  wave  had  completely  encircled  the  globe,  the 
average  velocity  was  684  miles  per  hour.  From  the  third  to  the  fifth  passage 
the  average  velocity  was  682  miles  per  hour.  From  the  fifth  to  the  seventh 
passage  the  average  velocity  was  676  miles  per  hour.  These  numbers 
are  taken  from  the  "Report  of  the  Royal  Commission  on  the  Eruption  of 
Krakatoa." 

The  Newtonian  velocity  at  o°  C.  is  about  640  miles  an  hour.  If  we 
: suppose  that  the  temperature  of  the  tropical  belt  was  25°  C.,  then  a  wave 
traversing  this  region  would  have  a  Newtonian  velocity  of  671  miles  per 
hour,  which  is  very  nearly  the  last  recorded  velocity.  The  Laplacian  value 
is  747  miles  for  o°  C.,  and  773  for  25°  C.  The  velocities  are  thus  very 
much  less  than  the  Laplacian  value  and  approximate  closely  to  the  New- 
tonian value.  There  is  every  reason  to  believe  that  the  succeeding  oscilla- 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE  99 

tions,  which  were  too  small  to  be  measured,  approached  the  Newtonian  value 
as  a  limit.  But  the  Newtonian  value  is  a  positive  limit.  No  disturbances 

IT 

in  a  gaseous  medium  can  be  propagated  with  a  less  velocity  than  \  /TS, 

while,  as  we  have  seen,  they  may  be  propagated  with  very  much  greater 
velocities  than  the  Laplacian  value. 

H.  J.  Rink  (1873),  in  "Poggendorf's  Annalen,"  Bd.  149,  pp.  533-546, 
was  the  first  to  call  attention  to  certain  differences  between  explosive  (uni- 
directional) waves  and  vibratory  (bidirectional)  waves.  He  pointed  out 
that  in  the  former  there  was  a  bodily  transference  of  the  gas,  which  remained 
in  its  new  position  all  along  the  line,  while  in  the  latter  each  particle  vibrated 
about  a  position  of  equilibrium,  and,  after  the  disturbance  had  passed, 
occupied  precisely  the  same  point  it  had  originally. 

Regnault  partly  recognized  this,  for  he  says :  "On  doit  done  admettre 
qu'au  moment  du  tir  d'une  arme  a  feu,  le  gaz  comprime  qui  s'en  echappe, 
est  lance  d'abord  comme  un  projectile,  qui  imprime  non  seulement  une 
compression,  mais  aussi  un  translation  aux  coudes  d'air  voisines.  Ce  dernier 
effet  devient  probablement  insensible  a  une  certaine  distance,  mais  il  doit 
troubler  notablement  la  vitesse  de  propagation  elastique  dans  le  voisinage 
du  depart.  J'ai  eu  souvent  occasion  de  reconnaitre  les  effets  de  translation 
dans  nos  experiences  surtout  dans  celles  qu'ont  ete  faites  dans  de  tuyaux 
de  petite  section." 

We  have  already  called  attention  to  the  fact  that  the  positive  and  nega- 
tive halves  of  a  vibratory  wave  have  trouble  in  keeping  step,  or  rather  in 
keeping  company.  The  natural  velocity  of  the  positive  half  is  greater  than 
that  of  the  negative  half,  so  that  the  former  is  continually  crowding  up  on 
the  latter.  A  compromise  is  at  first  effected  by  a  distortion  of  the  wave, 
the  crests  tilting  forwards  and  the  hollows  tilting  backwards.  The  profile 
of  the  wave  is,  therefore,  like  that  represented  in  Fig.  25. 

Now,  if  we  consider  a  particle  of  matter  imbedded  in  the  medium,  of 
a  density  exactly  equal  to  that  of  the  medium,  it  is  evident*  that  it  will  move 
to  and  fro  with  the  vibrations  exactly  as  if  it  were  a  part  of  the  medium. 
There  will  be  no  resultant  translation  of  the  particle  in  one  direction  or 
another.  The  case  is  very  different,  however,  if  the  particle  be  more  dense 
or  less  dense  than  the  medium.  Let  us  consider  the  case  of  a  denser 
particle.  As  the  wave  train  sweeps  over  it,  it  will  be  actuated  by  two 
forces,  viz.,  the  pressure  gradient  of  the  wave  and  the  streaming  past  it  of 
the  medium  first  in  one  direction  and  then  the  reverse.  As  the  compressed 
part  from  B  to  A  sweeps  by,  it  will  be  urged  to  the  right  by  an  excessive 
gradient  and  also  by  the  flow  to  the  right  of  the  condensed  medium.  From 
C  to  B,  the  medium,  of  less  density  than  normal,  will  stream  against  it  to  the 
left,  and  it  will  further  be  urged  to  the  left  by  a  more  gentle  pressure 
gradient.  But  the  important  factor  here  is  time.  The  compressed  half  will 


ioo  THE  ATMOSPHERE 

urge  it  to  the  right  for  an  appreciably  shorter  time  than  the  expanded  por- 
tion urges  it  to  the  left,  and  time  triumphs  in  this  case.  A  great  force 
acting  for  a  short  time,  as  the  blow  of  a  hammer,  will  move  a  body  less  than 
a  lesser  force  acting  for  a  longer  time,  as  the  push  of  a  man. 

For  a  discussion  of  this  problem,  the  reader  is  referred  to  the  article 
"Gravitation"  in  Popular  Astronomy  for  January,  1905.  It  can  be  shown 
that  a  denser  body  will  be  moved  against  the  wave,  while  a  less  dense  body 
will  be  carried  along  with  the  wave.  The  limiting  velocity  in  the  latter  case 
is,  of  course,  the  velocity  of  the  wave  itself. 

The  following  examples  will  serve  to  illustrate  this  peculiar  wave  action. 
If  a  balloon  filled  with  carbonic  acid  gas  be  brought  near  a  vibrating  tuning- 
fork,  it  will  move  towards  the  fork,  i.e.,  against  the  wave  train.  If  the 
balloon  be  filled  with  hydrogen  gas,  it  will  move  away  from  the  fork.  And 
if  the  balloon  be  of  the  same  density  as  the  air,  it  will  remain  stationary. 
The  author  believes  that  the  explanation  of  gravitation  is  closely  connected 
with  these  phenomena. 

Gravitation  is  not  merely  an  attractive  force  between  bodies ;  it  may  be 
attractive  or  repulsive  according  to  circumstances.  It  is  a  force  which  can- 
not act  through  nothing.  That  is,  it  must  produce  its  effects  through  the 
intervention  of  a  medium,  and  we  have  every  reason  to  believe  that  this 
medium  is  the  universal  ether.  Radiant  energy  not  only  sets  bodies  intp 
vibration,  that  is,  it  not  only  heats  them,  but  it  also  exerts  an  attractive 
force  on  bodies  denser  than  the  ether.  On  the  other  hand,  it  exerts  a  repul- 
sive force  on  bodies  less  dense  than  the  ether,  as  in  the  case  of  comets'  tails. 
The  sun,  by  the  waves  which  it  sends  through  the  ether,  drives  this  tenuous 
material  away  with  a  velocity  nearly  equaling,  or  perhaps  equaling,  that 
of  the  waves  themselves. 

Further,  the  ether  can  exert  no  attractive  or  repulsive  force  on  itself; 
that  is,  it  cannot  be  mutually  gravitative  or  repulsive.  It  is  simply  the 
bearer  of  waves.  It,  therefore,  of  itself  has  no  weight  or  temperature,  but 
it  confers  these  Attributes  on  other  bodies.  That  it  is  a  material  substance 
it  is  impossible  to  doubt,  since,  as  we  have  seen,  it  possesses  both  inertia 
and  elasticity,  the  fundamental  properties  of  all  material  substances.  It  is 
certainly  compressible,  else  it  could  not  transmit  waves.  It  differs  from 
ordinary  matter  chiefly  in  that  its  elasticity  is  very  great,  while  its  inertia 
is  very  small,  but  this  difference  is  merely  one  of  degree.  That  it  is  a 
uniform  continuous  body,  in  contradistinction  to  the  segregation  of  ordinary 
matter,  seems  highly  probable. 

OPTICAL  PHENOMENA 

THE    RAINBOW 

Let  Fig.  27  represent  a  magnified  raindrop,  and  A  0  the  direction 
in  which  the  sun's  rays  fall  on  it.  Some  of  these  rays  will  be  refracted,  suffer 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE 


101 


a  partial  internal  reflection  and  then 
be  refracted  out  again.  It  is  easily 
seen  from  the  figure  that  the  angle 
through  which  an  incident  ray  will 
have  been  turned  is  TT  -j-  2  i >  —  4  r. 
Now  as  the  parallel  rays  from  the 
sun  fall  on  different  points  of  the 
circumference  of  the  raindrop  it  is 
evident  that  the  angle  through 
which  each  is  turned,  or  the  total 
deviation  from  its  original  direc- 
tion, will  vary  with  the  angle  of  in- 
cidence. We  desire  to  determine 
the  particular  angle  of  incidence 
where  this  deviation  is  a  minimum. 

Differentiating  TT,  +  2  t  —  4  r  with  respect  to  i  and  equating  to  zero, 

dr 

we  find  that  -r-r  =  \  is  the  condition.     But  by  the  law  of  refraction  sin  i 
a  i 

dr      i  cos  i 

=  u  sin.  r,  where   u  is  the  index   of   retraction.     Hence,  -7— .  = 

d  i      }JL  cos  r 


FIG.  27 


cos 


i  — 


sin    t 


=  =  £.     The  index  of  refraction  //,  from  air  into  water,  is, 


for  the  middle  part  of  the  spectrum,  f.  Solving  our  equation,  we  find 
that  i—  59°  20'  and  that'the  minimum  deviation  is  138°. 

If  now  we  stand  with  our  back  to  the  sun  and  the  drop  is  moved  about 
in  a  circle  making  an  angle  of  42°  with  a  line  drawn  from  the  sun  and  pro- 
longed through  our  head,  it  is  evident  that  we  shall  see  the  ray  of  minimum 
deviation  at  all  points  of  this  circle.  If  we  move  the  drop  just  outside  this 
circle  it  is  evident  that  no  ray  will  now  reach  us,  since  the  deviation  of  all 
rays  will  be  greater  than  the  angle  between  our  line  and  the  drop. 

If  we  place  the  drop  anywhere  within  the  circle,  we  shall  receive  some 
rays  from  it  in  all  positions.  Even  near  the  center  of  the  circle  rays  will, 
theoretically  at  least,  be  reflected  directly  back  to  us.  Now,  in  considering 
all  the  rays  which  fall  upon  the  drop,  it  will  be  seen  that  a  great  many  will 
be  crowded  into  the  position  of  minimum  deviation,  for  as  we  increase  the 
angle  of  incidence  from  zero,  the  deviation  will  change  rapidly  with  the  angle 
until  we  approach  the  angle  requisite  for  a  minimum  deviation,  when  they 
will  mark  time ;  and  on  going  beyond  this  angle  they  will  retrace  their  steps. 
Hence,  if  we  distribute  drops  all  over  the  region  of  our  circle,  there  will  be 
a  very  faint  illumination  at  all  points  within  it ;  least  at  the  center  and  gradu- 
ally increasing  as  we  approach  the  circle ;  increasing  very  much  as  we  get 


IO2  THE  ATMOSPHERE 

near  the  circle ;  and  darkness  at  all  points  outside  the  circle.  If  now,  while 
the  sun  is  low,  a  shower  is  falling  on  the  opposite  side  of  the  horizon,  we 
shall  have  a  moving  picture.  Each  drop  on  entering  our  circle  will  for  a  mo- 
ment flash  out  brightly  and  then  become  practically  invisible.  It  is  easily 
seen  that  for  the  red,  the  least  refrangible  rays,  the  circle  will  be  largest, 
while  for  the  blue,  or  most  refrangible  rays,  the  circle  will  be  smallest. 
Hence,  our  circle  will  resolve  itself  into  a  spectrum  with  the  red  outermost 
and  the  blue  innermost.  The  circle  we  have  considered  is  called  the  first 
bow,  and  measurements  show  that  its  radius  is  42°,  as  we  have  found  ther 
oretically.  It  is  evident  that  this  first  bow  can  never  be  formed  when  the 
sun  is  higher  than  42°  above  the  horizon.  We  have  hitherto  considered 
only  the  .rays  striking  on  the  upper  half  of  the  drop.  Those  falling  on  the 

lower  half  form  a  precisely  similar 
but  reversed  primary  bow,  that  is, 
with  the  vertex  downwards,  which 
might  be  seen  by  a  man  in  a  balloon 
far  above  the  shower. 

We  saw  that  the  rays  forming 
the  first  bow  were  twice  refracted 
and  once  reflected,  but  on  striking 
the  inner  surface,  just  previous  to 

/-> ^  the  last  refraction,  all  the  rays  do 

/  not    emerge.     Some    are    reflected 

„  once  more  and  then  refracted,  and 

r  IG.    20 

theoretically  this  goes  on  indefi- 
nitely, but  practically  they  become  so  weakened  that  it  is  not  possible  to 
detect  any  bows  beyond  the  second,  that  is,  one  due  to  two  refractions  and 
two  reflections. 

By  the  same  reasoning  as  previously  applied,  we  find  that  this  second 
bow  is  concentric  with  the  first,  but  has  a  radius  of  51°. 

By  the  second  reflection  matters  are  reversed,  so  that  now  the  red  rays 
are  on  the  inside  and  the  blue  rays  on  the  outside.  Further,  the  faint  lumi- 
nosity that  we  found  within  the  first  circle  is  now  outside  the  second  circle, 
while  between  the  two  circles  there  are  absolutely  no  rays. 

If  the  light  forming  the  bows  came  from  a  point  and  was  homogeneous, 
our  circles  would  be  thin  rings,  but  as  the  sun  has  a  very  appreciable  disc 
and  the  light  is  not  homogeneous,  the  width  of  the  bows  is  considerably 
increased.  42°  2'  and  40°  17'  would  be  the  radii -of  the  respective  circles,  for 
the  first  bow  of  red  and  violet  rays  proceeding  from  a  point. 

Very  faint  supernumerary  bows  are  sometimes  seen  within  the  first  bow. 
We  saw  that  the  faint  rays  illuminating  the  inner  space  were  composed  of 
rays  which  had  suffered  a  greater  deviation  than  the  minimum.  For  every 
ray  on  one  side  of  the  minimum,  there  is  a  corresponding  ray  on  the  other 


OTHER  PHENOMENA  OCCURRING  IN  THE   ATMOSPHERE          103 

side  which  has  the  same  deviation,  and  hence  these  rays  interfere.  Some  of 
these  interfering  rays  have  a  half  wave  length  difference  of  phase,  others  a 
Wave  length,  and  so  on,  so  that  the  colored  rings  which  they  form  are  reg- 
ularly spaced.  They  are  usually  alternate  green  and  purple.  It  is  necessary 
for  their  production  that  the  raindrops  be  extremely  uniform  in  size.  The 
distance  between  the  supernumerary  bows  will  be  greater  as  the  drops  are 
smaller. 

There  are  many  disturbing  factors  to  prevent  the  formation  of  a  pure 
bow,  such  as  diffraction,  the  reflection  of  light  from  clouds  and  from  the 
drops  themselves,  etc.,  so  that  owing  to  overlapping  we  often  have  white 
bows. 

We  have  seen  that  cirrus  clouds  are  minute  ice  crystals  floating  in  the 
air  at  very  great  heights.  Ice  crystals  are  either  thin  plates  or  upright 
prisms,  but  the  predominant  angle  throughout  all  its  forms  is  60°.  They 
may  be  hexagons,  triangles  or  flower-shapes,  but  we  always  find  the  angles 
60°  and  90°.  The  refractive  index  of  ice  is  1.31,  and  the  refracting  angle 
is  always  60°  or  90°.  We  have  seen  that  refracted  rays  are  always  crowded 
together  in  the  direction  of  minimum  deviation.  Hence,  the  main  effect  of 
light  refracted  by  ice  crystals  in  the  air  will  be  seen  in  this  direction,  while 
the  straggling  rays  will  produce  a  much  feebler  illumination.  By  the  same 
method  we  have  used  for  rainbows,  we  find  that  light  issuing  from  a  celes- 
tial object  and  refracted  by  ice  prisms  of  60°  should  produce  a  circular  halo 
about  the  object  with  a  radius  of  22°.  Refracted  by  ice  prisms  of  90°,  the 
radius  of  the  halo  should  be  46°.  For  homogeneous  light  these  haloes 
would  have  a  width  about  equal  to  the  sun's  or  moon's  apparent  diameter. 
Measurements  of  the  ring  or  double  ring  sometimes  seen  about  the  sun  or 
moon  confirm  the  theory.  It  is  well  known  that  haloes  have  a  radius  of  22° 
or  46°,  and  they  are  spoken  of  as  22°  haloes  and  46°  haloes.  The  minimum 
deviation  is  least  for  the  red  rays ;  hence,  the  inner  border  of  the  haloes  will 
be  red  or  at  least  reddish.  For  the  same  reason,  the  straggling  rays  (rays 
not  of  minimum  deviation)  will  all  be  outside  of  the  ring.  The  inner  border, 
therefore,  will  be  more  sharply  delimited  and  its  color  more  perceptible  than 
that  of  the  outer  border.  The  external  portions  of  haloes  are  usually  prac- 
tically white. 

We  have  supposed  the  rays  to  pass  through  the  prisms  at  right  angled 
to  the  edges.  Much  of  the  refraction,  however,  will  be  oblique,  which  con- 
tributes to  destroy  definition.  When  the  crystals  in  falling  tend  to  set  in  a 
particular  direction,  edgewise  or  lengthwise,  and  the  sun  is  near  the  horizon, 
the  horizontally  refracted  rays  will  be  principally  seen  and  a  brilliant  image 
of  the  sun  may  be  perceived  at  the  same  level  to  the  right  or  left,  at  an  angle 
of  22°.  These  images  are  called  Parhelia  or  Mock-suns.  When  the  sun  is 
some  distance  above  the  horizon,  the  refraction  through  the  vertical  prisms 
will  be  oblique,  and  hence  the  angle  of  minimum  deviation  greater  than  22°. 


IO4  THE  ATMOSPHERE 

Hence,  as  the  sun  rises  higher  the  mock-suns  gradually  separate  outwards, 
still,  however,  keeping  the  same  apparent  altitude  as  the  sun.  If  the  prisms 
lie  mainly  horizontally,  the  mock-suns  will  be  directly  above  and  below  the 
sun,  provided  the  axes  of  the  prisms  are  perpendicular  to  the  line  joining  the 
sun  and  the  observer.  As  they  usually  lie  in  all  horizontal  directions,  the 
result  will  be  a  series  of  mock-suns  which  form  a  new  halo,  touching  the 
halo  of  22°  above  and  below  where  it  is  brightest,  and  gradually  fading  off 
on  each  side.  Such  haloes  are  called  tangent  arcs  to  the  halo  of  22°.  Of 
course  there  may  also  be  tangent  arcs  to  the  halo  of  46°.  We  have  sup- 
posed that  the  axes  of  the  prisms  are  mainly  directed  in  one  direction  in  fall- 
ing, the  vertical  or  the  horizontal,  and  this  is  probably  the  case  for  the 
brighter  parhelia  and  tangent  arcs,  but  it  is  not  necessary.  When  the  axes 
are  distributed  at  random,  the  selective  refraction  which  reaches  the  eye 
out  of  the  large  number  of  vertical  and  horizontal  prisms  may  give  rise 
to  the  phenomenon. 

The  subject  is  not  ended  here.  Sometimes  the  ice  prisms  are  terminated 
with  hexagonal  pyramids,  and  these  may  produce  haloes.  The  mock-suns 
themselves  may  produce  secondary  haloes,  so  that  the  results  may  be  very 
complicated.  Coronae  are  diffraction  phenomena  and  are  produced  when  the 
sun  or  moon  is  seen  through  a  mist  or  a  cloud.  They  are  concentric  colored 
rings  surrounding  these  bodies  and  are  equally  spaced.  The  width  of  each 
ring  depends  upon  the  size  of  the  drops  of  water,  which  must  be  very 
uniform.  The  smaller  the  drops,  the  wider  the  rings.  Being  diffraction 
rings,  the  red  is  on  the  outside,  instead  of  on  the  inside,  as  in  the  haloes. 


REFRACTION  DUE  TO  NON-HOMOGENEITY  OF  THE  AIR 

The  refractive  index  of  air  at  o°  C.  and  760  mm.  pressure  is  1.000294  or 
about  i  -f-  -gVoir-  Careful  measurements  show  that,  for  ordinary  ranges,  the 
excess  of  this  value  over  unity  is  proportional  to  the  density.  Or  the  refrac- 

4)  2  7  3  I 

tive  index  of  air  is  I  -f-  —-  •  — -^-~  • ( i ) ,  where  the  pressure  is 

760     273  +  /     3400 

expressed  in  mms.  of  mercury  and  the  temperature  is  centigrade.  Con- 
sidering a  small  portion  of  the  earth's  surface  a  plane  and  the  density  of  the 
air  equal  everywhere  at  the  same  level,  but  changing  with  the  level,  it  is 
evident  that  a  ray  of  light  will  travel  in  a  straight  line  from  one  point  to 
another  at  the  same  level.  A  ray  directed  obliquely  upwards,  as  it  passes 
through  layers  of  varying  density,  will  suffer  refraction,  and  as  the  densities 
change  continuously,  not  abruptly,  the  path  of  the  ray  will  be  a  curved  line. 
It  may  happen,  therefore,  that  one  ray  will  proceed  in  a  direct  line  from  one 
point  to  another  at  the  same  level,  and  that  another  ray  will  reach  it  by  a 
curved  line,  or  there  may  be  several  possible  curved  paths.  The  absolute 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE          105 

index  of  refraction  of  a  substance  is  the  ratio  of  the  velocity  of  light  in  a 
vacuum  to  its  velocity  in  the  substance. 

Since  we  can  resolve  an  infinitesimal  portion  of  the  path  of  a  ray  into  a 
horizontal  component,  which  does  not  change  its  velocity,  and  a  vertical  com- 
ponent which  does,  it  is  evident  that  the  acceleration  (or  retardation)  of  the 
ray  is  in  a  vertical  direction.  Denoting  the  vertical  direction  by  n,  we  have 


d  vn  _  d  vn     dn  _       d  vn  _ 

dt        dn      d  t~     "  dn  ~  dn 

since  v^,  the  horizontal  velocity,  does  not  change.  Hence,  the  acceleration  is 
measured  by  the  rate  of  change  of  the  kinetic  energy  per  unit  length  in  the 
direction  of  the  vertical.  We  say  kinetic  energy  because  we  suppose  a  small 
particle  to  represent  the  motion  of  the  ray.  It  is  evident  that  the  p'ath  must 
"be  symmetrical  with  respect  to  the  vertex,  which  is  half  way  between  two 
points  at  the  same  level,  from  the  principle  of  reversibility. 

Since  the  curvature  must  be  very  slight,  the  radius  of  curvature  has 
practically  a  vertical  direction  for  every  portion  of  the  path,  so  that  by 
ordinary  mechanical  principles  we  can  equate  the  centrifugal  force  to  the 
vertical  acceleration.  Now,  the  velocity  at  any  point  is  proportional  to  /f, 
the  index  of  refraction. 


TT  /**  \2/          1.  •      ^          j-  r  I         \      d  I*. 

Hence,  —  =  — -, — -.  where  p  is  the  radius  of  curvature,  or  —  =  —  •  -j— . 
'   p          dn   '  p      p    dn 

If  we  suppose  the  temperature  near  the  earth  to  be  constant,  say  o°  C, 
then  the  density  will  decrease  slightly  for  small  changes  of  level. 


By  (i) 


d  ft  _  dp          i 
d  n  ~  dn     3400  /0' 


34QO 


_  d_p_  i  dp_ 

dn       dn      3400/0  dn 

Under  normal  conditions  the  pressure  near  the  surface  of  the  earth 

decreases  -  -  for   each  meter,  since   the  height   of  a   homogeneous 
7997.8 

atmosphere  of  the  surface  density  would  be  7997.8  metres.     Thus,  -/- 

=  --      „.    Substituting  this  value  of  -J-  in  (2),  we  have   p  =  —  3401 
7997.8  dn 

X  7997-8-     (3) 


io6  THE  ATMOSPHERE 

The  minus  sign  shows  that  the  path  curves  downward.  The  curvature 
of  the  earth,  which  is  unity  divided  by  its  radius,  is  about  four  times  this 
amount.  If,  then,  we  introduce  the  curvature  of  the  earth,  we  see  that  it 
will  not  be  possible  for  a  ray  to  proceed  in  a  straight  line  between  two  points 
on  the  same  level,  but  one  point  -will  be  seen  from  the  other  by  means  of  the 
curved  ray  we  have  just  investigated.  For  this  reason  all  points  surrounding 
the  observer,  whether  on  the  horizon  or  above  it,  seem  to  be  raised  above  their 
proper  position.  The  position  of  the  point  directly  overhead  is,  of  course, 
not  changed.  In  ascending  in  a  balloon  likewise,  the  point  directly  under- 
neath is  judged  correctly,  but  the  rim  of  the  horizon  is  very  much  raised  and 
the  impression  conveyed  is  that  of  a  huge  bowl.  Balloonists  generally 
remark  that  the  earth  appears  to  be  hollowed  out  beneath  them. 


Since  u  =  i  -4-  —  •  — —  •  ,  if  the  temperature  changes  with  the 

A     273  +  /     3400 

dp  i       /        273  dp       p        273  d  t\ 

level,    we    have  -r-  = (7 .    ,.    .     •  -f-  —  -4-  -, r— r^  *  T~l- 

dn        3400   \(273  +  /)A      »»       A  (273  +  0        dnl 

i          i     /  i      dp        i       d  t\  _ 

Whence  -  = I  —  •  ~- •  -T—  I  very  nearly.      Or  we  can  write 

p      3400  \A     dn      273     dn) 

i  i      /    i  i     d  t\ 

very  nearly,  -  = ( H -j—  I. 

P          3400  \7997       273  dn) 

If  the  temperature  decreases  as  we  ascend,  then  -j—  is  negative  and 

further   if    -r—  = — ,  then  there  will  be  no   curvature   of  the  ray, 

dn  7997 

which  is  equivalent  to  saying  that  the  density  of  the  air  remains  constant. 
This,  under  normal  conditions  at  the  surface,  would  correspond  to  a  fall  of 
i  °  C.  for  about  every  30  meters.  The  real  average  rate  near  the  surface  is 

d  t  27  T. 

something  like  i°  for  every  200  meters.     If,  numerically,  -J—  >  — — ,  then 

the  curvature  of  the  ray  will  be  upward  and  we  shall  have  the  phenomenon 
of  the  mirage,  which  may  be  studied  by  turning  Fig.  29  upside  down. 
Objects  are  here  seen  depressed  below  the  horizon  and  give  the  observer  the 
impression  that  he  sees  them  by  reflection,  although  the  image  is  not  inverted 
as  it  would  be  by  reflexion.  However,  it  is  impossible  to  tell  whether  clouds 
are  inverted  or  not,  and  even  for  objects  which  might  be  tested  for  inversion, 
the  mind  does  not  stop  to  do  this,  but  judges  at  once  that  a  sheet  of  water 
lies  before  it.  Travelers  on  a  desert  where  the  air  is  intensely  heated  close  to 
the  surface  are  frequently  deceived  into  believing  that  water  lies  before  them. 
The  pressure,  no  matter  how  the  temperature  varies,  usually  changes  at 
a  practically  constant  rate.  If  the  temperature  also  changes  at  a  constant 
rate  with  the  height,  the  path  of  the  ray  will  be  an  arc  of  a  circle,  since  p 
will  be  constant.  Given,  therefore,  an  object  A  B,  Fig.  29,  and  knowing  the 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE 


107 


rate  of  change  of  temperature  with  the  height,  we  can  determine  p.  Then 
through  the  points  O  and  A  we  draw  a  circle  of  radius  p.  This  will  deter- 
mine the  path  of  a  ray  from  A  to  O.  Likewise,  the  same  circle  passing 
through  O  and  B  will  determine  the  path  of  a  ray  from  B  to  O.  Hence,  an 
observer  at  O  will  see  the  object  raised  and  erect.  This  is  the  case  of 
ordinary  atmospheric  refraction.  Since  the  rate  of  decrease  of  temperature 


FIG.  29 

with  height  is  generally  much  less  than  the  limiting  value  TyvV>  i-e->  smce 
the  air  in  general  becomes  less  dense  with  the  height,  all  objects  are  raised 
above  their  proper  positions.  In  navigation  and  astronomy  this  necessitates 
corrections  for  refraction.  While  these  corrections  may  be  applied  with  some 
degree  of  accuracy  when  the  body  has  a  considerable  altitude,  it  is  easily 
seen  that  when  the  body  is  near  the  horizon,  they  cannot  be  applied  with  any 
degree  of  trustworthiness. 

A' 


FIG.  30 

If  the  lower  air  has  a  constant  density  up  to  a  certain  level  L,  but 
beyond  this  a  constantly  decreasing  density,  it  will  be  seen  that  the  rays  will 
pursue  paths  as  indicated  in  Fig.  30.  In  the  lower  layer  the  paths  will  be 
straight,  while  above  they  will  be  curved  as  before.  The  path  of  the  ray 
from  the  lower  point  A  will  be  steeper  and  its  vertex  higher  than  that  of  the 
ray  from  B.  Hence,  their  paths  must  cross  and  the  image  will  be  inverted. 
Generally  inverted  images  are  magnified,  while  erect  ones  are  diminished 


io8  THE  ATMOSPHERE 

in  size.  Under  certain  conditions,  as  in  Fig.  30,  a  ship  may  be  seen  directly 
through  the  uniform  lower  layers  and  at  the  same  time  an  inverted  image 
of  it,  high  in  the  air  and  usually  magnified.  With  a  combination  of  layers, 
some  of  which  have  a  decreasing  density,  others  an  increasing  density,  while 
others  still  have  a  uniform  density,  the  different  possible  results  are  greatly 
increased.  Thus  a  ship  may  be  seen  directly  and  over  it  a  series  of  images, 
one  above  the  other,  some  inverted  and  some  erect,  in  every  possible  sequence. 
Objects  below  the  horizon,  which  under  normal  conditions  could  not  be 
seen  by  ordinary  refraction,  are  sometimes  raised  temporarily  by  these  abnor- 
mal refractions,  so  that  they  may  appear  high  in  the  air,  erect  or  inverted,  as 
the  case  may  be.  Sailors  call  this  "looming."  As  all  these  effects  are  due  to 
refraction,  the  images  have  colored  borders,  though  the  colors  are  not  par- 
ticularly noticeable. 

ELECTRICAL  PHENOMENA 

Since  we  do  not  know  what  electricity  is,  it  will  be  impossible  to  explain 
these  phenomena  in  the  complete  manner  in  which  other  phenomena  have 
been  explained.  We  must  content  ourselves  with  a  description  of  them  and 
the  various  surmises  which  have  been  advanced. 

The  atmosphere  always  contains  free  electricity,  which  in  the  vast 
majority  of  cases  is  positive,  and  increases  with  the  height  above  the  ground. 
The  difference  of  potential  may  be  as  much  as  600  volts  per  meter  (Exner). 
The  potential  appears  to  vary  inversely  as  the  vapor  pressure  and  conse- 
quently increases  with  cold.  Thus  certain  measurements  gave  a  potential 
of  325  for  2.3  mm.  of  vapor  pressure,  116  for  6.8  mm.,  and  68  for  12.5 
mm.  of  vapor  pressure.  In  fact,  the  electrification  of  the  atmosphere  seems 
to  depend  chiefly  upon  the  water  constituents,  vapor  and  ice.  Generally  the 
greatest  amount  of  electricity  is  observed  when  the  barometer  is  highest. 
Clouds  are  reservoirs  of  electricity  which  are  positive,  unless  by  inductive 
influence  from  a  more  strongly  charged  cloud  they  become  partly  negative. 
With  a  clear  sky  the  electrification  of  the  upper  regions  is  always  positive : 
if  negative  it  is  due  to  the  influence  of  clouds. 

The  potential  increases  with  condensation,  for  if  1000  vapor  particles 
each  possessing  the  same  charge  coalesce  to  form  a  single  droplet,  the  diam- 
eter of  the  droplet  will  be  only  ten  times  that  of  one  of  the  original  particles, 
and  since  the  capacity  is  equal  to  the  radius,  its  capacity  will  be  only  ten 
times  as  great.  But  the  charge  will  be  one  thousand  times  that  of  a  single 
particle ;  hence,  the  potential  will  be  one  hundred  times  as  much. 

The  electricity  of  the  ground  is  generally  negative  with  regard  to  the 
atmosphere,  but  a  cloud  induces  electricity  of  the  opposite  kind  on  the  earth's 
surface  directly  below  it.  Hence,  a  cause  of  earth  currents.  From  high 
exposed  points  electricity  is  always  passing  between  the  earth  and  the  atmos- 
phere. From  mountain  peaks  this  discharge  is  sometimes  luminous.  Lem- 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE          109 

strom  observed  in  winter  a  flame-like  appearance  from  the  tops  of  two  moun- 
tains, 800  and  noo  meters  high.  Occasionally  also,  clouds  which  are  dis- 
charging electricity  to  the  earth  become  luminous. 

The  cause  of  the  separation  of  the  two  electricities,  speaking  merely  in 
the  conventional  sense,  seems  to  be  due  to  friction  between  the  earth  and 
air  and  between  portions  of  the  air  itself  which  are  under  different  conditions. 
Faraday  showed  that  the  friction  between  minute  particles  of  water  and  dry 
air  is  an  abundant  source  of  electricity.  Armstrong's  hydro-electric  machine 
demonstrates  the  same  thing.  The  middle  and  polar  circulations  which  rub 
against  each  other  in  opposite  directions  at  their  border  should  be  a  source 
of  electricity. 

When  the  potential  between  two  clouds  or  a  cloud  and  the  earth  becomes 
great  enough,  the  insulation  will  be  broken  down  and  lightning  results. 
Flashes  of  lightning  are  often  a  mile  in  length  and  sometimes  as  much  as 
five  miles  in  length.  They  are  usually  zigzag,  as  after  proceeding  a  certain 
distance  in  one  direction,  the  resistance  increases  rapidly,  and  they  turn  in 
another  direction,  though  probably  not  as  sharply  as  it  appears.  The  electro- 
motive force  necessary  to  produce  a  spark  a  mile  long  in  air  at  normal  pres- 
sure has  been  estimated  as  something  over  3,000,000  volts.  However,  the 
droplets  are  intervening  conductors,  and  in  reality  the  lightning  does  not 
take  a  single  leap.  Hence,  it  is  possible  that  discharges  often  take  place  where 
there  is  no  very  great  difference  of  potential.  The  quantity  of  electricity, 
however,  is  always  very  great. 

There  are  several  varieties  of  lightning.  The  ordinary  zigzag  flash  is 
almost  always  between  earth  and  cloud  and  is  similar  to  the  discharge  of  a 
Leyden  jar.  Sheet  lightning  is  more  of  the  nature  of  a  brush  discharge 
between  two  clouds.  Ball  lightning  is  a  peculiar  form  which  is  not  yet  under- 
stood. It  consists  of  a  globe  of  fire,  from  an  inch  up  to  eighteen  inches  in 
diameter,  which  falls  to  the  earth  so  slowly  that  it  can  be  followed  by  the  eye. 
These  balls  often  rebound  on  striking  the  earth,  or  they  may  explode  with  a 
loud  report.  Plante,  with  an  enormous  battery  (great  quantity  and  tension), 
has  imitated  them.  He  used  as  electrodes  two  sheets  of  blotting  paper 
moistened  with  distilled  water,  which  is  a  very  bad  conductor.  Placing  these 
a  short  distance  apart,  he  was  able  to  produce  a  small  globe  of  fire  which 
rolled  about  between  them.  His  explanation  was  that  they  were  globes  of 
intensely  heated,  rarefied  and  dissociated  gases,  which  preserved  their  form 
by  means  of  an  envelope,  in  a  quasi-spheroidal  state,  which  was  practically 
non-heat-conducting.  The  electric  tension  between  earth  and  cloud  thus  pro- 
duces a  discharge  through  the  rarefied  globe  precisely  as  we  do  in  the  lab- 
oratory through  a  vacuum  tube.  The  fact  that  flashes  with  a  beaded  appear- 
ance, resembling  in  fact  a  rosary,  have  been  seen,  seems  to  bear  out  this 
explanation. 

St.  Elmo's  Fire — brush  discharges  from  the  ends  of  masts  and  spars  seen 


no  THE  ATMOSPHERE 

in  bad  weather  are  similar  to  the  discharges  we  have  already  spoken  of  from 
mountain  tops. 

Thunder  is  due  to  the  heating  and  tearing  effect  of  the  electricity  as  it 
passes.  Near  by  it  is  short  and  sharp ;  far  off  it  is  a  long  rumble,  due  to  the 
lengthening  and  flattening  of  the  sound  wave  with  distance.  Of  course,  to 
this  is  added  the  effect  of  echo  from  the  clouds  and  the  ground.  Thunder 
is  not  heard  at  very  great  distances,  the  energy  of  a  thunder  clap  not  being 
as  great  as  that  of  heavy  gun  fire.  The  cannonading  at  Waterloo  was  heard 
in  England.  Thunder  has  never  been  heard  at  such  a  distance,  fifteen  miles 
being  the  usual  limit. 

The  aurora  is  in  all  probability  a  brush  discharge  between  the  earth 
and  aqueous  particles  (ice)  in  the  higher  layers.  The  height  at  which  this 
phenomenon  takes  place  has  been  greatly  overestimated.  Balloons  have  oc- 
casionally been  in  the  midst  of  the  light  as  well  as  observers  at  the  surface 
of  the  earth.  The  following  is  taken  from  the  International  Cyclopaedia: 
"If  we  consider  the  aurora  as  a  discharge  through  aqueous  vapor  or  other 
gas,  then  we  have  nothing  to  do  with  the  gaseous  character  as  such,  but  with 
the  aqueous  component  only,  and  at  moderate  altitudes  the  density  of  the 
aqueous  vapor  is  so  slight  that  it  must  act  as  a  very  light  gas,  similar  to  that 
present  in  a  vacuum  chamber.  As  regards  the  height  of  the  aurora  above  the 
earth's  surface  several  methods  have  been  devised  for  calculating  it:  but 
all  trigonometrical  calculations  based  upon  most  careful  observations  seem 
to  show  that  the  definite  features  that  we  see  in  the  aurora  are  perspective 
phenomena  and  that  the  calculation  of  their  height  cannot  be  safely  made 
by  the  method  of  simultaneous  observations  at  two  stations.  In  fact,  the 
argument  for  the  existence  of  the  aurora  quite  close  to  the  earth's  surface  is 
too  strong  to  be  ignored." 

It  is  quite  possible  that  they  do  not  extend  much  above  seven  miles,  the 
region  of  aqueous  vapor. 

The  belt  of  greatest  frequency  of  auroras  has  been  found  to  be  a  curve 
passing  through  Point  Barrow,  the  northern  portion  of  Hudson's  Bay,  the 
southern  tip  of  Greenland,  Iceland,  the  North  Cape,  and  the  arctic  coast  of 
Russia  and  Siberia.  For  the  southern  aurora  there  is  also  such  a  circle  of 
maximum  frequency.  That  is,  observers  to  the  north  of  this  line  see  more 
auroras  to  the  south  than  to  the  north,  while  observers  to  the  south  see  more 
to  the  north.  This  line  corresponds  somewhat  with  the  border  between  the 
middle  and  polar  circulations,  where,  we  have  seen,  there  is  reason  to  believe 
that  an  excessive  amount  of  electricity  is  generated.  In  the  polar  regions, 
the  fall  of  potential  with  altitude  is  thirteen  times  greater  in  summer  and  eigh- 
teen times  greater  in  winter  than  at  the  equator.  Hence,  an  electrical  phe- 
nomenon, which  depends  upon  the  magnitude  of  this  fall  of  potential,  must  be 
more  intense  in  winter  and  high  latitudes  than  in  summer  and  in  the  torrid 
zones.  The  discharge  may  be  considered  as  passing  from  the  earth  to  the 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE          in 

aqueous  vapor  above,  and  as  not  becoming  luminous  (visible  to  the  eye)  until 
some  definite  altitude  (rarefaction)  is  reached.  The  rays  (currents)  have  a 
direction  in  general  identical  with  the  magnetic  lines  of  force  springing 
from  the  earth's  surface.  This  is  a  position  of  equilibrium  for  a  movable 
current.  It  is  well  known  that  a  current  capable  of  moving,  which  cuts  a  line 
of  magnetic  force,  will  be  moved  in  a  direction  at  right  angles  to.  both.  If 
the  ray  is  parallel  to  the  line  of  force,  it  will  not  be  moved,  except  perhaps 
twisted  spirally  about  the  line.  When  a  ray  moves  at  an  angle  with  the  line 
of  force,  it  will  be  displaced  and  the  quivering  usually  noticed  in  the  stream- 
ers is  probably  due  to  such  a  motion  imparted  by  the  magnetic  field  of  the 
earth. 

An  observer  looking  along  an  outgoing  ray,  i.e.,  end  on,  will  not  see 
anything.  Consequently  in  the  direction  of  the  lines  of  magnetic  force  there 
will  appear  overhead  a  less  illuminated  spot  in  the  heavens  which  is  called 
the  corona  of  the  aurora.  By  perspective  the  rays  from  the  outlying  points 
will  appear  to  converge  towards  this  point. 

If  there  are  ice  crystals  in  the  corona,  the  rays  of  light  reflected  nearly 
directly  backwards  will  at  some  points  interfere,  at  others  reinforce  them- 
selves, and  a  flower-shaped  or  rayed  figure  will  be  produced,  consisting  of 
alternating  dark  and  bright  bands  issuing  from  the  center.  The  edges  of 
these  bands  will  be  colored.  The  general  color  of  the  aurora,  as  its  name 
indicates,  is  reddish,  and  is  practically  the  same  as  that  produced  by  a  brush 
discharge  through  rarefied  nitrogen.  The  characteristic  yellow-green  line 
in  its  spectrum  is  due  to  krypton,  one  of  the  heaviest  gases  in  the  atmos- 
phere. 

The  rays  may  be  visible  at  the  surface  of  the  ground,  but  usually  there 
is  a  definite  interval  between  their  lower  extremities  and  the  ground.  This 
level  is  probably  dependent  upon  the  degree  of  rarefaction  necessary  for 
luminosity,  just  as  in  a  Geissler  tube  a  certain  degree  of  rarefaction  is  neces- 
sary before  luminous  effects  are  produced.  Thus,  it  is  probable  that  the 
greater  the  intensity  of  the  discharge,  the  nearer  will  the  streamers  appear 
to  be  to  the  earth,  and,  as  we  have  already  seen,  at  times  they  are  seen  spring- 
ing from  the  earth. 

The  structure  of  the  atmosphere,  that  is,  the  exact  arrangement  of  its 
different  layers  and  the  amount  of  its  vaporous  constituent  at  different 
points,  must  determine  at  any  instant  the  precise  configuration  of  the  aurora. 
The  upper  air  is  far  from  homogeneous  and  is  rather  to  be  characterized 
as  streaky. 

"A  remarkable  characteristic,  moreover,  met  with,  is  that  when  the  direc- 
tion of  such  wind  changes  the  change  may  be  perfectly  abrupt.  It  has,  indeed, 
been  recorded  by  scientific  balloonists  that  they  find  in  regions  where  winds 
of  different  directions  pass  that  one  appears  actually  to  drag  against  the 
surface  of  the  other,  as  though  tolerating  no  interval  of  calm  or  transition ; 


U2  THE  ATMOSPHERE 

and  yet  a  more  striking  fact  is  that  a  very  hurricane  may  brood  over  a  placid 
atmosphere  with  a  clear-cut  surface  of  demarcation  between  calm  and 
storm." — /.  M.  Bacon. 

Each  layer  is  moving  over  the  other,  so  that  the  conductivity  in  a  given 
direction  is  continually  changing,  and  the  appearance  of  an  aurora  may  be 
likened  to  an  image  which  is  projected  through  a  large  number  of  photo- 
graphic plates  which  are  continually  moving  relatively  to  each  other.  At  cer- 
tain points  in  the  lower  layers  the  rays  are  completely  cut  off,  while  at  other 
points  gaps  allow  them  to  pass  through,  just  as  gaps  in  a  cloud  let  beams  of 
the  sun  through.  The  aurora  is  thus  a  projection  or  picture  of  the  electrical 
conductivity  of  the  atmosphere.  It  has  frequently  been  remarked  that  the 
aurora  has  the  appearance  of  draperies  blowing  in  a  wind.  This  is  prob- 
ably an  actual  fact,  since  we  know  that  aloft  there  are  always  winds  blowing 
with  hurricane  velocity.  We  might  liken  the  atmosphere  to  a  gauze  curtain, 
which,  as  it  waves  in  the  wind,  is  lit  up  at  different  points  in  rapid  succession. 

It  has  been  noticed  that  the  aurora  moves  slowly  as  a  whole,  generally 
from  east  to  west,  but  sometimes  from  west  to  east.  It  seems  probable  that 
this  is  due  to  the  general  motions  of  the  polar  and  middle  circulations. 
When  the  aurora  is  on  the  polar  side  of  the  border  between  these  two  circula- 
tions, its  motion  should  be  towards  the  west.  It  would  seem,  therefore,, 
that  the  aurora  is  formed  more  frequently  in  the  polar  circulation. 

De  la  Rive  has  attempted  to  explain  this  phenomenon  by  an  ingenious- 
experiment  known  as  "De  la  Rive's  Experiment."  It  is,  in  fact,  an  applica- 
tion of  the  principle  that  a  movable  current  in  a  magnetic  field  is  thrust 
aside  in  a  direction  perpendicular  to  the  current  and  to  the  lines  of  magnetic 
force.  We  have  seen  from  the  foregoing  that  this  principle  may  explain  the 
quivering  of  the  rays,  but  it  cannot  explain  the  east-west  motion,  since  the 
rays  are  nearly  coincident  with  the  lines  of  force. 

An  aurora  is  usually  of  enormous  extent,  as  is  shown  by  the  magnetic 
perturbations  and  earth  currents  observed  coincidently  with  it  at  widely 
different  points,  even  where  it  may  not  be  visible.  The  luminosity  simply 
shows  that  where  this  occurs  the  vibrations  are  large  enough  to  affect  the 
eye.  Doubtless  if  the  eye  were  sensitive  enough  it  could  perceive  light  at  all 
points  where  the  streams  exist. 

Lastly  a  connection  undoubtedly  exists  between  auroras  and  sun-spots. 
The  frequency  of  auroras  increases  with  sun-spots,  there  being  an  eleven-year 
period  of  wax  and  wane  for  both.  It  would  seem  as  if  magnetic  lines  of 
force  were  shot  out  from  the  sun  along  the  axes  of  the  spot  vortices,  and 
these  lines  being  cut  by  a  moving  conductor — the  earth — produce  currents 
perpendicular  to  the  direction  of  motion,  i.e.,  in  a  north-south  direction. 
This  would,  of  course,  lead  to  brush  discharges  at  the  poles. 

The  auroral  discharge  frequently  produces  a  sound  like  the  rustling  of 
wind,  exactly  such  as  may  be  heard  in  the  brush  discharge  of  an  electrical 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE 

machine.     This  shows  that  the  observer  is  actually  in  the  midst  of  the  dis- 
charge, as  otherwise  it  could  not  be  heard. 

MECHANICAL  FLIGHT 

The  problem  of  mechanical  flight  naturally  resolves  itself  into  two  pos- 
sible methods — that  of  a  mechanism  which  has  the  same  or  about  the  same 
density  as  the  air,  and  therefore  floats  in  it,  and  that  of  a  mechanism  which 
is  heavier  than  air  and  is  kept  afloat  by  its  own  energy,  which  it  expends 
against  the  inertia  of  the  air  in  a  downward  direction.  In  other  words,  it 
drives  the  air  downwards,  thereby  overcoming  its  inertia,  and  the  reaction 
from  overcoming  this  inertia  of  the  air  urges  it  upwards.  Since  this  reaction 
is  equal  to  the  mass  of  air  displaced  into  its  acceleration,  or  the  rate  at  which 
its  downward  velocity  is  increased,  it  follows  that  by  applying  sudden  and 
powerful  thrusts  against  the  air,  a  heavy  body  may  be  supported  or  even 
raised. 

Of  the  first  form  of  airship,  it  need  only  be  said  that  in  still  air  a  bal- 
loon, provided  that  its  volume  be  kept  constant,  floats  at  some  definite  level, 
which  is  where  its  density  is  equal  to  that  of  the  surrounding  air.  By 
expending  energy  in  thrusts  against  the  inertia  of  the  air  in  a  horizontal 
direction,  it  will  move  the  air  in  that  direction,  and  will  move  itself  in  an 
opposite  direction,  and  the  total  mass  of  the  air  moved  into  the  distance  it. 
has  moved  will  be  equal  to  the  mass  of  the  balloon  into  the  distance  it  has 
moved.  For  such  a  dirigible  balloon,  of  course,  it  will  be  necessary  to  have 
an  elongated  or  cigar  shape  in  order  to  reduce  to  a  minimum  the  resistance 
of  the  air  in  front  of  it.  Generally,  unless  it  is  moving  with  the  wind  and 
with  the  exact  velocity  of  the  wind,  it  will  have  to  overcome  a  virtual  wind, 
which  is  the  geometric  difference  between  the  velocity  of  the  actual  wind  and 
the  velocity  of  the  balloon.  In  order  to  overcome  the  pressure  of  this  virtual 
wind  it  will  have  to  possess  some  rigidity,  as  otherwise  it  would  be  crushed. 
We  have  seen  that  the  wind  pressure  on  a  plane  surface  opposed  to  it  nor- 
mally was  proportional  to  the  area  of  the  surface  into  the  square  of  the 
velocity  of  the  wind.  The  formula,  P=  .003  V*  S,  expresses  the  pressure 
in  pounds  per  square  foot  when  the  velocity  is  given  in  miles  per  hour.  The 
pressure  increases  rapidly  with  the  velocity.  Hence,  it  would  be  impossible 
for  a  dirigible  to  go  against  a  very  strong  wind,  even  provided  it  had 
unlimited  power.  We  have  seen  that  a  hurricane  blowing  eighty  to  one  hun- 
dred miles  an  hour  often  destroys  permanent  buildings  of  the  most  solid 
construction.  No  dirigible,  therefore,  could  withstand  such  exceptional  pres- 
sures. It  could,  however,  go  with  a  gale,  without  any  great  trouble,  only  a 
certain  amount  of  airmanship  being  necessary  to  overcome  the  vertical  com- 
ponents. 

Dynamically,  the  aeroplane  is  more  interesting.     If  a  plane  be  moved 


H4  THE  ATMOSPHERE 

horizontally  through  the  air  with  a  slight  angle  of  elevation,  it  will  continu- 
ally thrust  down  a  certain  mass  of  air,  which  by  its  reaction  will  tend  to  pre- 
vent the  plane  from  falling.  To  support  the  plane  without  rising  or  falling, 
the  energy  expended  will  have  to  be  precisely  that  necessary  to  thrust  down 
a  mass  of  air,  such  that  its  reaction  will  be  equal  to  the  weight  of  the  plane. 
Thus  to  keep  an  aeroplane  afloat  requires  the  constant  expenditure  of  energy, 
and  a  considerable  amount  at  that. 

It  has  been  supposed  that  birds  can  soar  without  the  expenditure  of  any 
energy,  and  that  it  might  be  possible  to  construct  an  aeroplane  which,  once 
started,  would  maintain  itself  in  the  air  without  the  further  expenditure  of 
any  energy.  The  idea  is,  of  course,  erroneous,  as  all  birds  are  continually 
falling.  A  leaf  upheld  by  a  vertical  current  is  really  falling  through  this  air 
and  at  a  considerable  velocity.  The  resistance  to  its  fall  through  this  air  is 
equal  to  its  weight,  and  hence  relatively  to  the  earth  it  appears  stationary. 
So,  when  we  see  a  bird  remaining  at  the  same  level,  practically  without  mov- 
ing a  feather,  we  may  be  sure  that  it  is  falling  relatively  to  the  air  around  it, 
or,  in  other  words,  it  is  supported  by  a  vertical  current  of  which  it  is  taking 
momentary  advantage. 

So,  too,  an  aeroplane  is  continually  falling,  and  it  falls  through  a  dis- 
tance equal  to  the  vertical  distance  between  its  front  and  rear  edges,  in  the 
time  it  takes  for  the  rear  edge  to  move  up  to  where  the  front  edge  was.  It 
is  the  resistance  to  this  fall  which  supports  it.  When  the  engine  of  an  aero- 
plane is  stopped,  it  glides  on  to  be  sure,  but  the  energy  now  expended  in  dis- 
placing the  air  downwards  is  drawn  from  its  kinetic  energy,  and  as  the  veloc- 
ity decreases,  this  is  not  sufficient  to  keep  it  aloft  and  it  sinks  towards  the 
earth. 

We  have  seen  that  the  thrust  between  the  aeroplane  and  the  air  is  equal 
to  the  mass  of  air  displaced  into  its  acceleration  at  any  instant.  The  air  is 
not  only  pushed  down  by  the  under  surface,  but  it  is  also  pulled  down  by 
the  upper  surface.  At  the  front  edge,  where  the  air  is  dead,  this  change  of 
velocity,  starting  from  zero,  is  most  sudden,  and  therefore  the  reaction  great- 
est. At  the  rear  edge,  since  the  air  is  now  moving  rapidly  downwards,  the 
acceleration  is  least.  If  the  velocity  of  the  air  here  is  exactly  equal  to  the 
rate  at  which  the  surface  is  moving  downwards  (along  its  incline),  there  will 
be  no  reaction  upwards.  If  the  velocity  of  the  air  downwards  is  greater  than 
this,  then  there  will  be  a  negative  reaction,  or  this  air  will  pull  the  rear  edge 
downwards.  Hence,  beyond  a  certain  point  it  is  a  disadvantage  to  make  the 
plane  too  wide,  as,  outside  of  the  increased  weight,  we  may  convert  a  thrust 
upwards  into  a  thrust  downwards.  We  see  then  that  the  thrust  against  the 
under  surface  is  not  the  same  at  every  point,  but  is  greatest  at  the  front  edge 
and  least  at  the  rear  edge,  so  that  the  center  of  pressure  does  not  correspond 
to  the  center  of  surface  or  center  of  gravity  of  the  plane,  being  in  advance 
of  the  latter.  The  result  is  that  a  couple  is  always  at  work  tending  to  bring 


OTHER  PHENOMENA  OCCURRING  IN  THE   ATMOSPHERE  115 

the  front  edge  up  and  the  whole  plane  at  right  angles  to  the  direction  of 
motion. 

In  a  lateral  direction,  however,  we  gain  a  distinct  advantage  in  increas- 
ing the  edge  of  the  plane,  since  the  lifting  thrust  is  directly  proportional  to 
the  length  of  this  edge.  Since  the  front  edge  engages  only  dead  air,  the  limit 
of  the  length  of  this  edge  can  only  be  set  by  consideration  of  practical  con- 
struction, rigidity,  etc.  Large  birds  have  great  length  of  wing  from  tip  to 
tip,  but  very  little  width.  If  it  were  not  for  this  principle,  we  may  be  sure 
they  would  have  been  made  with  short,  broad  wings.  An  advantage  is 
gained  in  curving  the  supporting  surface  so  that  the  rear  edge  shoots  the  air 
directly  downwards.  The  front  edge  then  grips  dead  air,  and  this  grip  will 
be  more  or  less  maintained  by  the  curve.  Birds'  wings  are  all  curved. 

Thus  we  see  that  it  is  necessary  to  construct  aeroplanes  of  great  length 
laterally,  but  small  width  fore  and  aft.  However,  as  too  great  an  extension 
laterally  is  inconvenient,  if  not  impracticable,  the  most  advantageous  construc- 
tion is  probably  a  series  of  steps  sloping  backwards,  so  that  we  obtain  a  great 
length  of  front  edge  for  cutting  dead  air.  By  this  arrangement  each  plane 
engages  practically  undisturbed  air,  which  would  not  be  the  case  if  the 
planes  were  vertically  over  each  other. 

We  have  seen  that  the  maximum  thrust  comes  upon  the  front  edge 
where  the  air  is  dead,  and  that  a  couple  always  exists  tending  to  turn  the 
plane  flatwise  to  the  wind.  A  card  always  turns  in  falling,  so  as  to  bring 
its  surface  perpendicular  to  the  line  of  fall.  It  balances  for  an  instant  in 
this  position,  when  it  slides  off  again  in  another  direction,  but  shortly  brings 
up  by  presenting  its  surface  again,  and  thus  by  a  series  of  swaying  move- 
ments, between  which  it  comes  nearly  to  rest,  gradually  falls  to  the  ground. 
The  center  of  pressure  moves  quickly  and  seemingly  erratically  over  the 
plane,  but  actually  depends  upon  the  velocity  and  angle  of  inclination. 
It  would,  for  this  reason,  be  clearly  impossible  to  construct  an  aeroplane  with 
a  single  plane,  since,  however  we  might  ballast  it  fore  and  aft,  it  would  be 
in  unstable  equilibrium.  A  slight  increase  or  decrease  in  the  velocity  would 
tilt  it  up  or  down. 

A  great  deal  of  trouble  has  been  taken  to  calculate  mathematically  the 
center  of  pressure  for  a  given  velocity,  angle  of  elevation,  shape  of  plane, 
etc.,  but  all  this  has  little  practical  bearing.  Very  slight  changes  in  any  of 
these  quantities  will  cause  a  considerable  and  in  practice  unforseeable  shift- 
ing of  the  center  of  pressure.  The  problem  of  fore  and  aft  equilibrium  can 
only  be  solved  by  the  use  of  two  planes,  well  separated  in  a  fore  and  aft 
direction.  It  does  not  matter  then  at  what  point  the  pressure  comes  in  each 
plane.  There  is  a  forward  pressure  and  a  rear  pressure,  and  by  regulating 
one  of  these  by  a  lever,  equilibrium  can  always  be  maintained.  Birds  effect 
this  equilibrium  by  means  of  their  head,  and  legs  and  tail,  chiefly  the  latter. 

Lateral  equilibrium  is  effected  in  much  the  same  way.     By  raising  the 


u6  THE  ATMOSPHERE 

tip  of  one  plane  and  depressing  the  other,  increased  pressure  is  brought  to 
bear  on  one  side  and  decreased  pressure  on  the  other.  This  couple,  acting 
through  a  long  arm,  turns  the  aeroplane  laterally,  so  that  equilibrium  can 
always  be  effected  by  the  aeronaut.  Birds  secure  lateral  equilibrium  in  the 
same  manner.  By  the  turn  of  the  tip  of  a  wing  they  right  themselves.  Since 
they  have  the  power  of  moving  the  long  wing  feathers  separately,  this  is 
easily  accomplished. 

It  seems  difficult  to  set  limits  to  the  possible  development  of  the  aero- 
plane in  the  future.  By  increasing  the  number  of  planes  and  setting  them  at 
different  levels,  preferably  sloping  backwards,  so  that  in  case  of  accident  they 
might  break  the  fall  by  opposing  a  maximum  surface  to  a  vertical  descent, 
a  single  airship  is  practically  multiplied  into  several  as  regards  lifting  power. 
This  would  mean  that  many  passengers  might  be  carried  instead  of  one.  The 
power,  however,  would  have  to  be  multiplied  in  like  proportion,  and  this 
brings  us  to  the  present  immediate  difficulty — the  motor. 

Gasoline  engines  are  now  made  weighing  only  a  few  pounds  per  horse- 
power, but  it  seems  unlikely  that  this  will  be  the  motor  of  the  future.  As 
the  power  increases,  the  weight  of  a  gas  engine,  at  best,  increases  in  a  rather 
greater  proportion ;  they  are  far  from  reliable  and  are  lacking  in  flexibility, 
especially  as  the  size  increases.  It  would  seem  preferable  that  the  energy 
should  be  stored  in  a  separate  reservoir  instead  of  being  producd  in  the  cyl- 
inders. Steel  flasks  are  now  made  of  no  great  weight  capable  of  withstand- 
ing a  pressure  of  two  thousand  pounds  to  the  square  inch.  If  the  explosions 
were  produced  in  such  a  reservoir,  the  necessity  for  compressing  the  gas 
previous  to  explosion  with  the  accompanying  loss  of  stroke  and  energy  would 
be  avoided.  The  necessity  of  cooling  the  cylinders  would  also  be  done  away 
with  and  the  weight  of  the  water  used  in  cooling  would  be  saved.  This 
stored  energy  could  be  used  in  a  turbine  or  reciprocating  engine.  Recipro- 
cating engines  have  been  constructed  weighing  less  than  a  pound  per  horse- 
power. With  improvements  in  metallurgy  there  is  no  reason  to  doubt  but 
that  some  alloy  may  be  found  for  the  blades  of  turbines  capable  of  withstand- 
ing the  action  of  very  hot  gases.  The  improvement  of  motors,  therefore., 
lies  in  the  concentration  of  very  great  energy  in  a  very  small  weight. 

What  we  have  said  about  planes  applies  to  propellers,  for  propellers  are 
planes.  They  must  be  long  and  narrow  and  a  slight  curve  is  advantageous 
so  as  to  increase  the  grip  of  the  posterior  half.  Since  the  velocity  is  pro- 
portional to  the  distance  from  the  center,  the  inner  surface  may  be  dispensed 
with  as  not  producing  an  effect  commensurate  with  its  weight. 

The  helicopter,  where  the  planes  of  a  propeller  are  used  as  the  sup- 
porting planes,  is  impracticable  in  its  ordinary  form.  By  this  method  an 
inverted  cyclone  is  produced  in  the  air.  Hence,  the  air  after  a  few  turns  is 
moving  down  with  the  surface  which  seeks  to  gripe  it.  There  is  no  dead 


OTHER  PHENOMENA  OCCURRING  IN  THE  ATMOSPHERE 


117 


air  on  which  to  obtain  a  grip,  except  at  the  very  beginning-.     Hence,  the 

somewhat  paradoxical  result  has  been  found  experimentally  that  the  lifting 

power  does  not  increase  with  the 

velocity  of  the  propeller   and   the 

energy    expended,    but    may    even 

decrease. 

By  shielding  the  upper  surface 
of  the  screw,  as  in  Fig.  31,  the  air 
entering  from  the  sides  only  in  a 
horizontal  direction  may  be  sud- 
denly shot  down  and  thus  becomes  practically  dead  air,  for  the  upward  thrust 
is  equal  to  the  mass  of  air  thrust  down  into  the  acceleration  of  this  thrust. 

The  general  principle  holds  that  the  upward  thrust  on  the  aeroplane  is 
proportional  to  the  square  of  the  difference  between  the  velocities  of  the 
plane  and  the  air.  Hence,  an  aeroplane  cannot  sail  with  the  wind  and  with 
the  same  velocity  as  the  wind.  It  is  a  matter  of  relativity.  When  an 
aeroplane  sails  with  and  against  currents  of  varying  velocity,  the  thrust  given 
by  the  propellers  will  have  to  be  varied ;  all  of  which  shows  the  advantage 
of  great  reserve  power  and  flexibility. 


APPENDIX 


FIG.  32 


THE  GYROSCOPE 

In  Fig.  32  let  us  suppose  that  G  I  is  a  disc  capable  of  spinning  about 
the  axis  0  G,  which  passes  through  its  center  G.  Further  the  axis  W  O  G 

is  capable  of  turning  about  the  fixed  point 
O  in  any  direction.  We  can  abolish  gravity 
by  fixing  a  counterbalancing  weight  W  at 
the  other  end  of  the  axis,  so  that  the  system 
will  move  exactly  as  if  gravity  did  not  exist. 
We  shall  first  give  the  disc  a  spin  in  a 
counterclockwise  direction,  so  that  the  point 
/  on  the  disc  will  be  coming  up  through 
the  page.  Now,  let  us  hit  the  system  a 
smart  tap,  so  that  it  will  tend  to  revolve 
about  the  point  O  directly  downwards 
through  the  page.  That  is,  it  will  start 
to  revolve  about  the  axis  O  P,  which  is  per- 
pendicular to  O  G.  It  is  now  left  to  itself, 

no  further  forces  acting  on  it.  and  the  problem  is  to  determine  the  subse- 
quent motion. 

We  have  here  a  double  rotation,  and  according  to  our  definition,  it  is 
this  which  constitutes  a  gyroscope.  The  points  on  the  disc  to  the  left  of  G 
are  coming  up  through  the  page,  and  the  whole  system  is  moving  bodily 
downwards  through  the  page.  Now,  there  must  be  some  point  to  the  left  of 
G  where  these  two  opposite  velocities  exactly  counterbalance  each  other. 
Let  us  suppose  that  the  point  7  on  the  disc  is  this  particular  point.  For  the 
first  instant  of  motion,  therefore,  the  system  will  move  as  if  it  had  only  a 
simple  rotation  about  the  axis  O  I.  We  shall  call  this  axis  the  Instantaneous 
Axis.  We  shall  call  the  constant  angular  velocity  with  which  the  disc  is 
spinning  (&,  and  we  shall  suppose  that  the  blow  imparted  to  the  system  an 
angular  velocity  /*  about  the  axis  O  P. 

Since  the  point  7  remains  for  the  first  instant  in  the  page  and  does  move 
during  this  instant,  we  have 

GI(*>=OGn.     (i) 

We  do  not  say  that  the  system  will  continue  to  rotate  about  the  axis  O  I, 
for  such  is  not  the  case.  We  only  say  that  at  the  first  instant  it  will  start  to 
do  so.  Now,  the  moment  of  momentum  of  the  disc  about  the  axis  O  G  is 


APPENDIX  119 

M  h*  co,  where  M  represents  its  mass  and  k  is  its  radius  of  gyration.  This 
simply  means  that  the  quantity  of  motion  in  the  disc  would  be  the  same  if 
all  its  mass  were  concentrated  into  a  thin  ring  at  the  distance  k  from  the 
center,  and  this  ring  rotated  with  the  same  angular  velocity  GO.  The  moment 
of  this  quantity  about  the  axis  O  G  is  found  by  multiplying  it  by  k. 

Now,  it  is  evident  that  the  moment  of  momentum  about  an  axis  is  a 
quantity  which  can  be  resolved  into  a  component  about  any  other  axis  by 
multiplying  it  by  the  cosine  of  the  angle  between  the  two  axes.  Let  us  draw 
through  the  point  O  two  lines  O  X  and  O  L  mutually  perpendicular.  Drop 
a  perpendicular  from  G  to  p  on  O  L,  and  call  the  angles  G  0  I  and  G  O  L, 
i  and  y  respectively.  Let  the  angular  velocity  of  G  about  O  L  be  Q  and 
about  O  /,  GO{. 

Now,  the  point  G  is  passing  through  the  page  with  a  velocity  0  G  sin 
i  &i,  and  the  disc  is  at  the  same  time  rotating  about  this  point  G,  its  center 
of  gravity,  with  angular  velocity  GO. 

Then  its  moment  of  momentum  about  the  line  O  X  is  M  •  O  G*  sin  i. 
cos  Y  wi  —  M  k*  GO  sin  y.  (2) 

The  first  term  represents  the  moment  of  momentum  of  the  mass,  sup- 
posed to  be  concentrated  at  its  center  of  gravity,  about  the  axis  O  X,  and  the 
second  term  is  the  component  about  this  axis  due  to  the  spin.  They  are 
given  opposite  signs  because  they  rotate  in  opposite  directions  about  O  X. 

It  is  clear  that  the  moment  of  momentum  about  O  X  will  vary  as  we 
change  the  direction  of  this  line.  But  we  can  always  find  one  direction,  and 
one  only,  in  the  plane  G  O  I,  such  that  this  quantity  will  vanish.  We  have 
only  to  choose  the  angle  y  so  that  the  two  partial  moments  we  have  just 
found  shall  be  equal.  In  other  words,  the  condition  is  that 

j/ .  Q  G*  •  cos  y  •  sin  i  •  GO{  =  M  •  k*  •  sin  y  •  GO.     (3) 

We  see  then  that  there  will  be  absolutely  no  turning  movement  about 
such  a  line  O  X.  In  other  words,  the  point  G  will  start  off  in  a  direction 
parallel  to  the  plane  drawn  through  O  X  and  perpendicular  to  O  L.  But 
once  started  in  this  direction,  since  no  further  forces  act  upon  it,  it  must  for- 
ever afterwards  move  parallel  to  this  plane. 

Since  the  point  G  must  remain  at  a  constant  height  above  this  plane,  it 
follows  that  the  axis  O  G  will  maintain  a  constant  angle  with  the  line  O  L; 
in  fact,  describe  a  cone  about  it.  This  line  O  L  we  shall  call  the  Invariable 
Line.  It  is  evident  that  the  instantaneous  axis  O  I  will  always  remain  in  the 
plane  G  O  L,  so  that  the  subsequent  motion  of  our  gyroscope  is  completely 
determined.  It  would  be  the  same  as  if  we  imagined  a  cone  G  O  I  fixed  to  the 
gyroscope  rolling  upon  a  cone  G  O  L  fixed  in  space.  This  is  the  motion  of  a 
gyroscope  started  off  with  an  initial  impulse  and  then  left  to  move  without 
the  action  of  any  further  forces.  The  axis  O  G  will  continue  revolving  about 


T2O 


THE  ATMOSPHERE 


the  invariable  line  0  L  at  a  constant  angle  y  and  with  a  constant  angular 
velocity  ft.  The  system,  as  a  whole,  will  at  any  instant  be  turning  about  an 
instantaneous  axis  O  I  in  the  plane  G  O  L,  and  this  axis  O  I  will  make  a 
constant  angle  i  with  the  axis  O  G. 

Let  us  denote  the  moment  of  inertia  of  the  disc  about  the  axis  O  P,  or 
M-  O  G*  by  A,  and  its  moment  of  inertia  about  the  axis  O  G  by  C.  From 
(3)  we  have,  then,  A  sin  i  cos  y  oot  =  C  GO  sin  y.  Now,  &?=(»,•  cos  i; 
hence,  A  tan  i  =  C  tan  y.  Knowing,  therefore,  the  mass  of  the  gyroscope 
and  the  initial  velocities  of  the  spin  and  turn,  the  whole  subsequent 
motion  of  the  system  is  determined. 

We  can,  instead  of  representing  the  motion  as  the  rolling  of  one  cone 
upon  another,  equally  well  represent  it  by  supposing  a  surface  of  revolution 
to  be  attached  to  the  gyroscope  and  allowing  this  surface  to  roll  upon  a  plane 

perpendicular  to  O  L  and  tangent  to  the 
surface.  The  point  of  contact  between  the 
surface  and  the  plane  will  trace  two  circles, 
one  on  the  surface,  the  other  on  the  plane. 
We  shall,  however,  have  to  select  our  sur- 
face so  that  the  angular  velocity  of  the  point 
of  contact  in  the  circle  on  the  plane  will  be 
ft,  while  its  angular  velocity  in  the  circle  on 
the  surface  will  be  GO.  Poinsot,  who  first 
investigated  this  motion,  called  the  circle  on 
the  surface  the  Polhode,  and  the  circle  on 
the  plane  the  Herpolhode.* 

Let  us  now  draw  a  line  G  L,  Fig.  33, 
perpendicular  to  0  G.  This  line  will  evi- 
dently describe  a  cone  tangent  to  a  sphere 
drawn  about  O  wtih  the  radius  O  G.  The 


FIG.  33 


path  of  G  will  be  a  circle  on  this  sphere.  Since  the  gyroscope  is  revolving 
at  a  constant  rate  and  distance  from  the  point  L,  it  follows  that  the  centri- 
fugal and  centripetal  forces  directed  away  from  and  to  this  point  must  be 
•equal.  The  centrifugal  force  tending  to  throw  G  away  from  L  is 

M-  O  G  -sin  y  cos  y  £1*.  Hence,  in  the  motion  we  have  considered,  a 
force  must  be  developed  acting  in  the  direction  G  L  and  equal  to  M-  OG- 
sin  y-cos  y  fl3.  This  force  we  shall  call  the  gyroscopic  force. 

Since  A    tan    i  =  C  tan   y   and   O.  sin    y  =  GO  tan  z,  we  may  write 


_ 
M-  O  G  •  sin  Y  cos  y  fl2  = 


_  .  _ 
O  G 


k1  sin  y  £1  GO. 


But  D.  sin  y  is  the  angular  velocity    with  which  the  center  G  is 
turning  in  its  path  about  the  fixed  point  O.    In  other  words,  at  any  instant 
*  Poinsot.     Theorie  nouvelle  de  la  rotation  des  corps,  1834  and  1852. 


APPENDIX  121 

the  gyroscopic  force  is  always  directed  at  right  angles  to  the  plane  of  motion 
of  the  gyroscope,  towards  the  pole.  Further,  this  gyroscopic  force  is  equal 
to  the  mass  into  the  square  of  the  radius  of  gyration,  into  the  angular  velocity 
of  the  spin,  into  the  angular  velocity  with  which  it  is  turning  in  its  path 
about  its  fixed  point,  divided  by  the  distance  of  its  center  of  gravity  from 
the  fixed  point.  This  is  the  fundamental  property  of  a  gyroscope  from  which 
all  others  can  be  easily  derived.  We  shall  now  be  able  to  understand  all  the 
peculiarities  of  motion  of  such  a  body  under  all  circumstances. 

To  recapitulate,  then,  we  have  found  this  to  be  the  fundamental  property 
of  a  gyroscope,  viz.,  that  if  we  turn  a  spinning  disc  about  a  fixed  point  in  its 
axis  in  any  direction  (or  plane),  there  will  immediately  be  set  up  a  force 
pulling  upon  it  in  a  direction  at  right  angles  to  the  direction  in  which  it  is 
moving.  If  the  disc  were  not  spinning,  its  path  would  be  in  the  direction  of 
the  force  moving  it,  but  the  spinning  sets  up  a  force  at  right  angles  to  its 
path  at  any  instant.  We  have  seen  that  if  we  start  it  off  in  any  direction 
and  then  allow  no  further  outside  forces  to  act  upon  it,  it  will  continue  to 
circle  about  a  fixed  line  0  L,  and  that  the  natural  deflecting  force  is  continu- 
ally pulling  it  away  from  the  great  circle  it  otherwise  would  describe  on  the 
sphere.  A  small  spin  would  only  deflect  it  slightly,  so  that  it  would  describe  a 
circle  nearly  as  large  as  the  great  circle  it  otherwise  would  describe,  and  tan- 
gent to  this  great  circle  at  the  point  of  starting. 

From  the  previous  discussion,  we  see  that  with  a  very  great  spin, 
*  and  y  both  become  very  small,  and  therefore  the  result  of  striking  a 
heavy  blow  on  a  rapidly  spinning  gyroscope  would  be  to  cause  it  to  oscillate 
rapidly  around  a  very  small  circle  passing  through  its  initial  position.  Here 
we  find  to  a  certain  extent  a  confirmation  of  the  popular  idea  that  a  gyro- 
scope tends  to  preserve  its  plane  of  rotation.  But  a  gyroscope  has  no  ten- 
dency to  preserve  its  plane  of  rotation.  It  changes  its  plane  of  rotation  read- 
ily, only  in  a  manner  peculiar  to  itself.  It  is  rather  startling  to  find  that  with 
an  infinitely  great  spin,  we  might  strike  our  gyro- 
scope a  heavy  blow  with  a  hammer  and  yet  be 
unable  to  move  it.  But  this  is  merely  a  limiting 
case,  and  the  proper  way  to  regard  it  is  that  a 
gyroscope  offers  no  resistance  to  a  change  of 
plane,  but  changes  it  in  the  manner  we  have  just 
investigated  and  now  understand. 

We  come  now  to  consider  the  result  of  a 
constant  and  constantly  directed  force,  such  as 
gravity,  upon  a  spinning  body. 

Let  #,  Fig.  34,  be  the  angle  which  the  axis  FIG.  34 

0  G   makes   at   any    instant   with   the    opposite 

direction  to  gravity,  and  ^  the  angle  which  a  vertical  plane  through  O  and 
£  makes  with  the  initial  vertical  plane.     The  angles  #  and  ty  express  the 


122 


THE  ATMOSPHERE 


colatitude  and  longitude  of  the  point  G  at  any  instant.  We  shall  represent 
the  distance  O  G  by  /,  and  suppose  the  disc  spinning  in  a  counterclockwise 
direction.  Taking  the  south  and  east  directions  as  positive,  we  can  write 
from  the  fundamental  principle  of  the  gyroscope. 

jn  7j 

/  sin  #  ip  —  — —  $  (i)  and  g  sin  ft  —  /  $  =  — j-  ip  sin  -9.        (2) 


,a  tan  — 

Integrating  (i)  we  have  ^  =  — j^-  log — , 

7  tan-0 

2 

where  #0  is  the  initial  colatitude. 

Substituting  (3)  in  (2),  we  have 


(3) 


sin  *  —  /  *  = 


Multiplying  by  #  and  integrating, 


sn 


—  g  cos  9 = 

2 


a 


log  sin  #  —  cos  #  log 


7? 

tan  — 


If  the  gyroscope  starts  from  rest,  K  =  —  g  cos  #0  — 
Hence, 


log  sin  #0. 


—  =  —  =  /  g  (cos  #0  —  cos 
2  2 


sin 


0 

log  —  —  r2  +  cos  *  log 
s  sm  t?  6 


tan  — 

2 

tan  ^ 

2   . 


,      (4) 


where  ^,  denotes  the  polar  velocity. 

I  g   (cos  &o  —  cos  »9)  represents  the  work  done  on  the  gyroscope  by 

2  /  Z.2     _  _\  1 

V  .  IK    OJ\ 

gravity,  while— —  is  the  kinetic  energy  of  the  polar  velocity,  and  I — T— 1 

#  — i 
tan  — 

sin  w                               2 
l°g  ~  ~~s cos  ^  1°S o~       ^s  ^^e  ^metic  energy  of  the  horizontal 

2  — 

velocity.  Besides  the  initial  value  #0  one  other  value  of  '9  will  cause  v^ 
to  vanish.  The  vertical  movement  of  the  gyroscope  is,  therefore,  con- 
fined between  these  limiting  values. 

The  gyroscope  then  at  first  begins  to  fall  vertically,  but  immediately  a 
horizontal  motion  is  superadded.  By  the  principle  of  energy,  the  total  veloc- 
ity at  any  instant  will  be  proportional  to  the  square  root  of  the  vertical  drop* 


APPENDIX 


123 


but  this  is  distributed  between  the  vertical  and  horizontal  components  in  re- 
spectively decreasing  and  increasing  proportions,  until  finally  the  motion  is 
all  horizontal.  After  this  it  begins  to  rise,  until  at  the  original  level  it  comes 
to  rest  again,  and  the  process  is  repeated  over  and  over  again. 

We  shall  next  investigate  the  path  which  the  point  G  traces.  In  order 
to  simplify  matters  we  shall  consider  only  a  very  small  portion  of  the  sphere 
which  may  be  taken  as  practically  a  plane.  Hence,  1$  and  /  x  will  be  rec- 
tangular coordinates  within  its  limits,  where  x  —  ^  sm  a-  In  this  small 
area  we  can  consider  the  moment  of  gravity  constant  and  equal  to  Ig  sin  a. 
Our  equations  of  motion  now  assume  the  following  simplified  forms  : 


co 


,  .  ••  • 

(?  =  —  -j  —  (5)  and  g  sin  a  —  I  *  =  —  -j-  ip.     (6) 

Jn  Jn 

Integrating,  we  have  I  $  —  —j-  $     (7)      and  g  sin  a  t  —  I  ft  —  —j-  </>  (8), 

since    we    assume    the    origin    of    coordinates    to    be    where    the    body 
starts  from  rest. 

If  we  write  the  equations 

1*  =  (Kt-smKt),         (9) 


„ 

where  K  = 


/*  =  _     (l_Cos  A-/),  (10) 

k*  GO 


we  can  easily  see,  by  differentiating  and  substituting  the  values  for  sin  K  t 
and  cosKt,  that  (9)  and  (10)  are  the  integrals  of  equations  (7)  and  (8). 
Now  equations  (9)  and  (10)  represent  a  cycloid  with  its  base  horizontal, 
convex  downwards,  and  having  a  cusp  at  the  origin  of  coordinates.  Of 
course  the  point  G  cannot  leave  the  sphere  and  a  cycloid  is  a  plane  curve, 
but  in  the  small  part  of  a  sphere  which  we  can  consider  to  be  plane,  it  traces 
a  part  of  a  cycloid.  Every  portion  of  the  path  is,  therefore,  a  portion  of 
some  cycloid  in  a  plane  tangent  to  that  particular  point  of  the  sphere.  It 
follows  that  the  point  G  in  tracing  its  path  upon  the  sphere  moves  in  the 
quickest  time  possible  under  the  circumstances  from  one  point  of  its  path  to 
another.  For  the  cycloid  is  the  curve  of  quickest  rise  and  fall,  and  from  any 
one  point  to  another  near  it  the  gyroscope  moves  in  the  shortest  time  possible. 
The  equations  of  a  cycloid  are  usually  written  x  —  a  (q>  —  sin  (p) 
and  y  =  a  (i  —  cos  q>),  where  a  is  the  radius  of  the  generating  circle  and  q> 
is  the  angle  which  a  radius  makes  at  any  time  with  its  initial  position.  The 
radius  of  our  generating  circle,  and  therefore  the  cycloid,  is  quite  large  if  <*> 
be  small,  but  if  GO  be  large  the  generating  circle  becomes  very  minute.  With 
a  rapid  spin,  therefore,  it  would  be  possible  to  trace  a  large  number  of 
cycloids  even  in  the  small  area  to  which  we  have  restricted  ourselves. 


124  THE  ATMOSPHERE 

The  radius  of  the  generating  circle  is  g  ^f,  or  g  Sm4  a  /*.      The 

J\.  K     GO 

time  of  falling  through  one  of  these  cycloids  is  found    by  making  Kt 

,  .       2    7t  2    7f  r 

equal  to  2  TT,  and  therefore  is  -7^-  or 


-^  --  . 
K  K  co 

We  see  that  the  time  of  one  of  these  vibrations  is  independent  of  the  actuat- 
ing force  or  g. 

We  now  have  a  complete  knowledge  of  the  motion  of  our  gyroscope.  If 
the  spin  be  small,  there  is  a  considerable  rise  and  fall  which  is  easily 
observed.  It  here  traces  out  a  path  which  we  might  call  a  spherical  cycloid, 
because  every  element  of  it  is  part  of  a  definite  cycloid,  which  could  be 
drawn  upon  the  plane  tangent  to  the  sphere  at  that  element.  With  a  very 
rapid  spin,  the  rise  and  fall  cannot  be  observed  by  the  eye,  though  the  ear 
•can  take  up  the  humming  caused  by  the  minute  vibrations.  The  motion  here 
is  the  same  as  if  the  extremity  of  the  axis  were  attached  to  the  circumference 
of  a  very  minute  wheel  and  this  wheel  were  rolled  along  on  the  under  side  of 
a  parallel  of  latitude.  The  axis,  therefore,  carves  out  in  space  a  cycloidally 
fluted  cone  with  the  cusps  upward. 

If  at  the  instant  of  starting  we  could  give  the  gyroscope  an  initial  hori- 
zontal velocity,  such  that  its  upward  deflecting  force  was  exactly  equal  to 
the  component  of  gravity,  then  there  would  be  no  rise  and  fall,  and  the  axis 
would  describe  a  smooth  cone.  This  would  be  equally  the  case  with  a  small 
or  a  great  spin.  If  the  spinning  ceases,  the  cycloid  degenerates  into  a  great 
vertical  circle  and  the  motion  becomes  simply  that  of  a  pendulum  passing 
at  every  swing  through  the  point  vertically  under  the  point  of  support.  But 
it  will  never  pass  through  this  point  as  long  as  it  possesses  the  slightest  spin. 

We  now  see  clearly  why  a  'gyroscope,  or  top,  supports  itself  in  seeming 
defiance  of  gravitation.  As  soon  as  it  acquires  a  vertical  angular  velocity 
downward,  normal  deflecting  forces  are  set  up  which  pull  it  away  from  this 
direction,  and  the  direction  of  its  path  being  continually  changed,  it  is  event- 
ually pulled  up  to  the  level  from  which  it  started.  This  is  repeated  in  rapid 
oscillations  over  and  over  again.  If  we  constrained  the  gyroscope  to  move 
in  a  vertical  plane,  as  by  placing  the  extremity  of  the  axis  in  a  vertical  groove, 
it  would  swing  like  a  pendulum,  exactly  as  if  it  were  not  spinning,  provided 
we  do  not  take  into  account  the  friction  caused  by  the  horizontal  pressure. 

We  have  seen  that  the  fundamental  property  of  a  gyroscope  is  that 

—  -  —  $  =  Mf,  where  /represents  an  acceleration  perpendicular  to  the 

O  G 

plane  of  #.     This  means  that  the  normal   turning   moment  about  the 
point  O  is  M  k*  GO  #. 

Now,  if  we  turn  a  gyroscope  about  its  own  center  of  gravity,  O  G  will 
become  zero  and  }  mathematically  infinite.  But  the  product  /.  O  G  still 


APPENDIX  125 

remains  finite,  and  is  the  turning  moment  of  a  couple  exerted  in  a  plane  per- 
pendicular to  the  plane  of  5,  or  the  plane  in  which  we  turn  it,  and  passing 
through  the  axis.  This  moment  is  numerically  equal  to  M  k*  GO  #. 

We  have  seen  that  a  gyroscope  progresses  a  distance  equal  to  the 

2  n  g  sin  a  . 
circumference  of  the  generating  circle  of  its  cycloid,  or  -    -^-^ ,  in  the 

2    71  2   U  k*   GO    . 

time  —f^-.     It   will,    therefore,    require   a   time   equal  to -. —  for  a 

J\.  £ 

complete  revolution,  which  is  independent  of  the  inclination  of  its  axis. 


SOME  MEMORABLE  HURRICANES 


The  first  experience  of  Europeans  with  a  West  Indian  hurricane  was 
in  the  latter  part  of  1495.  Columbus  was  at  that  time  at  the  little  settlement 
of  Isabella,  on  the  north  coast  of  San  Domingo,  making"  preparations  to 
return  to  Spain.  "When  the  ships  were  ready  to  depart  a  terrible  storm 
swept  the  island ;  it  was  one  of  those  awful  whirlwinds  which  occasionally 
rage  within  the  tropics,  and  which  were  called  'uricans'  by  the  Indians,  a 
name  which  they  still  retain.  Three  of  the  ships  at  anchor  in  the  harbor 
were  sunk  by  it,  with  all  who  were  on  board ;  others  were  dashed  against  each 
other,  and  driven  mere  wrecks  upon  the  shore.  The  Indians  were  over- 
whelmed with  astonishment  and  dismay,  for  never  in  their  memory  or  in 
the  traditions  of  their  ancestors  had  they  known  so  tremendous  a  storm."* 

The  Great  Storm  of  November  25,  1703  devastated  nearly  the  whole 
of  England.  "Fifteen  sail  of  the  line,  with  Admiral  Bowater  and  all  his 
crews,  with  several  hundred  merchantmen  were  lost.  London  appeared  like 
a  city  which  had  sustained  a  protracted  siege,  whole  streets  being  destroyed, 
and  several  thousand  individuals  buried  beneath  the  ruins.  Eight  thousand 
seamen  perished."  An  annual  sermon  is  still  preached  in  London  on  Novem- 
ber 25th,  in  commemoration. 

The  hurricane  of  August  31,  1772  destroyed  ten  thousand  lives  at 
Martinique. 

The  hurricane  of  October  10,  1780  called  "The  Great  Hurricane,"  was 
of  great  extent  and  extraordinary  violence,  as  is  shown  by  its  rapid  northing. 
It  proceeded  from  Trinidad,  by  way  of  Santa  Lucia,  to  Bermuda.  A  large 
number  of  British  and  French  men-of-war  foundered.  The  French  gov- 
ernor of  Martinique  made  the  laconic -report,  "The  ships  disappeared." 

The  hurricane  of  April  I,  1782  destroyed  all  the  prizes  taken  by  Rod- 
ney, together  with  an  immense  number  of  merchantmen  and  nearly  all  the 
men-of-war  convoying  the  fleet. 

The  hurricane  of  September  16,  1782  is  known  as  Admiral  Graves'  dis- 
aster. "The  cyclone  continued  at  N.  W.,  and  before  it  left  the  hapless  fleet, 
the  whole  of  the  men-of-war,  except  the  Canada,  including  the  flagship,  and 
all  the  merchantmen,  had  foundered.  So  large  was  the  proportion  of  mer- 


*  Washington  Irving. 


SOME  MEMORABLE  HURRICANES  127 

-chantmen  that  this  is  supposed  to  be  the  greatest  naval  disaster  on  record. 
Upwards  of  three  thousand  seamen  alone  are  computed  to  have  perished." 

The  Atlantic  coast  storm  of  March  n,  1888,  which  came  up  from  the 
Bahamas,  and  which  is  known  as  "The  Great  Blizzard,"  destroyed  nearly  all 
the  coast  shipping  at  sea. 

The  great  Samoan  hurricane  of  March  16,  1889  wrecked  and  stranded 
six  men-of-war  and  eight  merchantmen,  with  a  large  loss  of  life. 

The  Porto  Rican  hurricane  of  August  8,  1899  destroyed  three  thou- 
sand lives.  (Fig.  35.) 

The  Galveston  hurricane  of  September  8,  1900  destroyed  the  city  of 
Galveston,  and  with  it  six  thousand  lives. 

The  inundation  wave  accompanying  the  tremendous  cyclone  of  June, 
1822  destroyed,  at  Burisal  and  Backergunge  at  the  mouths  of  the  Burram- 
pooter  and  Ganges,  upwards  of  fifty  thousand  souls. 


128 


THE  ATMOSPHERE 


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FIG.  35. — Barograph  of  Porto  Rican  Hurricane  of  August  9,  1899,  taken  at  San  Juan, 
by  MR.  J.  A.  CANALS,  Civil  Engineer. 


SOME  MEMORABLE  HURRICANES 


129 


VELOCITY  OF  THE  EARTH  AT  VARIOUS  LATITUDES 
MILES  PER  HOUR 


Lat. 

Naut. 

Stat. 

Lat. 

Naut. 

Stat. 

0° 

900 

1037-5 

31 

773 

890 

i 

32 

764.7 

780.6 

2 

33 

756-3 

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3 

34 

747-7 

861 

4 

35 

738.7 

850.8 

5 

897.6 

1033  6 

36 

729.7 

840.3 

6 

37 

720.4 

829.6 

7 

38 

710.8 

818.5 

8 

39 

701.  i 

807.3 

9 

40 

691 

795-9 

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887.4 

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680.8 

784.2 

ii 

884.5 

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42 

670.5 

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12 

881.4 

1015 

43 

660 

760 

r3 

878 

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44 

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747.6 

14 

874.3 

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45 

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734-8 

15 

870.4 

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46 

627 

721.9 

16 

866.2 

997.6 

47 

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17 

861.7 

992.4 

48 

603.9 

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18 

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592.2 

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981.3 

5° 

580.2 

668.2 

20 

847 

975-3 

5' 

568.2 

654-3 

21 

841.5 

969 

52 

556 

640 

22 

835.8 

962.4 

53 

543-3 

625 

23 

829.8 

955-5 

54 

53°-7 

61  1.  1 

24 

823^5 

948.3 

55 

5*7-9 

596.4 

25 

817 

940.8 

56 

5°4-9 

581.4 

26 

810.3 

933 

57 

491.8 

566.4 

27 

803.2 

925 

58 

478.5 

55I-1 

28 

796 

916.8 

59 

465.1 

535-6 

29 

788.5 

908 

60 

451.6 

520 

3° 

781 

899.2 

OF  THE 

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